ads:
Robust Ergodic Properties in Partially
Hyperbolic Dynamics
Martin Andersson
Tese apresentada ao Instituto Nacional de Matem´atica Pura e Aplicada,
em Julho de 2007, para a obten¸c˜ao do grau de Doutor em Matem´atica
Orientada por Marcelo Viana
Abstract
We study ergodic properties of partially hyperbolic systems whose
central direction is mostly contracting. Earlier work of Bonatti, Viana
[BV] about existence and finitude of physical measures is extended
to the case of local diffeomorphisms. Moreover, we prove that such
systems constitute a C
2
-open set in which statistical stability is a
dense property. In contrast, all mostly contracting systems are shown
to be stable under small random perturbations.
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This work has been carried out with financial support from CNPq.
Contents
1 Introduction 6
2 Some preliminary notions and description of results 8
2.1 Partial Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Mostly contracting central direction . . . . . . . . . . . . . . . 9
2.3 Robustness and Statistical stability . . . . . . . . . . . . . . . 10
2.4 Stable ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Some related problems . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Stochastic stability . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Toolbox 14
3.1 Integral representation of measures . . . . . . . . . . . . . . . 14
3.2 Admissible measures and carriers . . . . . . . . . . . . . . . . 15
3.2.1 Admissible manifolds . . . . . . . . . . . . . . . . . . . 15
3.2.2 The Grassmannian bundle . . . . . . . . . . . . . . . . 17
3.2.3 Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.4 Simple admissible measures . . . . . . . . . . . . . . . 19
3.2.5 An interlude into the heuristics of admissible measures 20
3.2.6 Topology on A and K . . . . . . . . . . . . . . . . . . 20
3.2.7 Admissible Measures . . . . . . . . . . . . . . . . . . . 24
3.2.8 A disintegration technique . . . . . . . . . . . . . . . . 25
3.2.9 Ergodic admissible measures . . . . . . . . . . . . . . . 33
3.2.10 Generic Carriers . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Stable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Absolute continuity . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Finitude of physical measures and the no holes property 37
5 Robustness and statistical stability 39
5.1 A characterisation of the mostly contracting hypothesis . . . . 39
5.2 Semi-continuity of Lyapunov exponents . . . . . . . . . . . . . 40
5.3 Large stable manifolds . . . . . . . . . . . . . . . . . . . . . . 41
5.4 The natural extension and balanced lifts . . . . . . . . . . . . 45
5.5 Statistical stability . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Stochastic stability 49
5
1 Introduction
A sound approach to understanding smooth dynamical systems consists of
giving a statistical description of most orbits. It is sensible due to the extreme
complexity of the orbit structures, so frequently encountered in dynamical
systems with some expanding behaviour. In practice this often boils down to
finding out whether a given system f has a physical measure, i.e. a probability
measure µ for which the basin
B(µ) := {x ∈ M :
1
n
n−1
k=0
δ
f
k
(x)
weakly
→ µ} (1)
has positive Lebesgue measure. Successful work on Axiom A diffeomorphisms
[Ru, Si, Y] has lead dynamiscists to believe that many dynamical systems
can be satisfactorily described on a statistical basis — a view taken by Palis
in his well-known conjecture on the denseness of finitude of attractors [P].
Such a description would encapsulate topics like
Existence: There are physical measures for the system.
Finitude: The number of physical measures is finite.
No holes: Lebesgue almost every point in the manifold M belongs to the
basin of some physical measure.
Statistical stability: All physical measures persist under s mall perturba-
tions.
Stochastic stability: Physical measures describe random orbits of the sys-
tem under small noise.
Since the seventies, physical measures have been proved to exist in much
greater generality than Axiom A diffeomorphisms, including some partially
hyperbolic systems [BV, ABV, T], the H´enon family [BY], and others.
In the present work, we study an open set of C
2
partially hyperbolic local
diffeomorphisms f : M → M on compact Riemannian manifolds, mostly
contracting along the central direction. Such systems provide a non-invertible
generalization of mostly contracting diffeomorphisns, first studied by Bonatti,
Viana [BV], and later by Castro [C] and Dolgopyat [D]; however this time
the focus is on statistical stability. Particularily under the possibility of
coexistence of several physical measures on the same attractor.
A conceivable obstacle to statistical stability is the seemingly pathological
phenomenon, present in a fascinating example due to Kan [K], exhibiting two
6
physical measures supported in the same transitive piece of the dynamics. It
seems likely that this phenomenon can be destroyed by small perturbations
of the system, thus leading to a bifurcation in the set of physical measures.
Kan’s example falls into a class of systems which we nowadays call partially
hyperbolic with mostly contracting central direction. It is known from the
work of [BV] that if the unstable foliation is minimal for a mostly contracting
diffeomorphism, then there is only one physical measure. To what extent this
occurs is not known, although some research has been made on the subject
[BDU, PuSa], suggesting it to be a common feature.
Nevertheless, the present work introduces a new set of techniques to deal
with statistical (and stochastic) stability of mostly contracting systems, in-
dependently of whether they exhibit Kan’s phenomenon or not. We prove:
• Mostly contracting contracting local diffeomorphisms have a finite num-
ber of physical measures and satisfy the no holes property.
• Having mostly contracting central direction is a robust property.
• The number of physical measures vary semi-continuously with the dy-
namics.
• Sytems that do not alter the number of physical measures under small
perturbations are statistically stable.
• These make up an open and dense subset of all mostly contracting
systems.
• In particular, all systems with a unique physical measure are statisti-
cally stable.
• Among mostly contracting conservative diffeomorphisms, every ergodic
system is necessarily stably e rgodic.
• All mostly contracting systems are stochastically stable.
A key feature of the arguments used is that they apply to non-invertible
maps just as well as diffeomorphisms, provided that there are no critical
points. This is done by replacing the traditional Gibbs-u states [PeSi] with
a multi-dimensional analogue of Tsujii’s admissible measures [T]. The cur-
rent approach is even more advantageous in the non-invertible case, where
uniqueness of the physical measure is harder to obtain due to the lack of
unstable foliation.
7
Acknowledgements
This b eing my first independent work in mathematical research, I would like
to seize the oportunity to thank all those people that have tought me math-
ematics; particularily those who have put trust into my academic progress.
I am refering here to Stefano Luzatto, who introduced me to dynamical sys-
tems, and Marcelo Viana, for accepting me as his student at IMPA where I
have learnt most of what I know in the field. I also thank Fl´avio Abdenur for
acting as an encouraging force and for being such a fierce promoter of semi-
continuity arguments. My fellow student Yang Jiagang owes a great thank
for pointing out Corollary D and its proof, as do all other students at IMPA,
with whom I have exchanged ideas on a daily basis. My last mention goes
to Augusta for providing such a divine Bobˆo de Camar˜ao — it has certainly
had a good effect on my work!
2 Some preliminary notions and description
of results
Let M a smooth compact Riemannian manifold. To avoid trivial statements,
we will suppose the dimension to be at least two. Denote by Diff
2
loc
(M) the
space of C
2
local diffeomorphisms on M, i.e. C
2
maps whose derivative
is of full rank at every point. It is an open subspace of C
2
(M, M) and,
in particular, contains all diffeomorphisms. Elements of Diff
2
loc
(M) will be
referred to as systems, or simply maps.
We deviate slightly from standard terminology and say that Λ is an at-
tractor for the system f if Λ is a compact f-invariant set and there exists an
open neighbourhood U of Λ, called a trapping region, such that
f(U) ⊂ U and Λ =
n≥0
f
n
(U).
In other words, there is no requirement of transitivity and, in particular, M
itself is always an attractor with trapping region M.
2.1 Partial Hyperbolicity
Several notions of partial hyperbolicity may currently be found in the litera-
ture, of which the most widely known requires a decomposition of the tangent
bundle into three complementary subbundles (see [AV] for discussion). The
type cinsidered in this work requires only two complementary subbundles,
one of which is uniformely expanded under the action of the system and
8
dominating the other. It is usually referred to as partial hyperbolicity of
type E
u
⊕ E
cs
.
Thus an attractor Λ is partially hyperbolic under f if there exists a split-
ting T
Λ
M = E
c
⊕ E
u
into non-trivial subspaces, a constant 0 < τ < 1, and
an integer n
0
such that
(Df
n
|
E
u
x
)
−1
≤ τ
n−n
0
(2)
Df
n
|
E
c
x
(Df
n
|
E
u
x
)
−1
≤ τ
n−n
0
(3)
both hold for every x ∈ Λ and every n ≥ 0.
The subspace E
c
x
above is necessarily unique, and varies continuously with
x. On the other hand, E
u
x
is not. In fact, when f is non-invertible, there is
typically no invariant unstable direction at all. Still, we can always define a
strictly invariant conefield
S
u
x
= {v
c
⊕ v
u
∈ E
c
x
⊕ E
u
x
: v
u
≥ αv
c
}
for some α > 0. Strict invariance here means that Df
x
S
u
x
is contained in
the interior of S
u
f(x)
for every x ∈ U. The subspace E
c
x
is characterised by
those vectors v ∈ T
x
M such that Df
n
x
v /∈ S
u
f
n
(x)
for every n ≥ 0. There is no
harm in supposing that E
u
is smooth, say C
∞
. The lack of invariance of E
u
is reflected in the following observation: Let . . . x
−2
, x
−1
, x
0
, . . . y
−2
, y
−1
, y
0
be two different pre-orbits of a point x
0
= y
0
. Then
n≥0
Df
n
S
u
x
−n
is not
necessarily the same as
n≥0
Df
n
S
u
y
−n
.
Upon possibly replacing U by a subset, and slightly altering the constants
n
0
, τ, we may suppose that the splitting and unstable cone field extend to
the whole of U, and (2), (3) hold for every x ∈ U.
We denote by PH(U, S
u
) those f ∈ Diff
2
loc
(U) that leave U and S
u
strictly
invariant and admit a partially hyperbolic splitting satisfying (2), (3) for some
τ < 1. It is an open subset of Diff
2
loc
(M).
We call E
c
the central direction of f, and use the notation
D
c
f := Df
|E
c
in all that follows. The letters c and u will also denote the dimensions of E
c
and E
u
— the central and unstable dimensions.
2.2 Mostly contracting central direction
The maximum central Lyapunov exponent is the map
λ
c
+
: PH(U, S
u
) × U → R
(f, x) → lim sup
n→∞
1
n
log D
c
f
n
(x).
9
We rephrase the definition of mostly contracting diffeomorphisms used in
[BV] suitably into our context.
Definition 1. A system f ∈ PH(U, S
u
) is mostly contracting along the
central direction if, given any disc D ⊂ U (at least C
1+Lip
) tangent to S
u
,
there exists a subset A ⊂ D of positive Lebesgue measure such that λ
c
+
(f, x) <
0 for every x ∈ A.
After characterising this definition in Section 5.1, it will become clear
that it coincides with that of [BV] in the case of diffeomorphisms. The space
of mostly contracting systems in PH(U, S
u
) will be denoted by MC(U, S
u
).
We shall be irresponsible and omit explicit mentioning of the trapping re-
gion and unstable conefield. Thus when saying that f is partially hyperbolic
(f ∈ PH), it is understood that there exists some trapping region U and
a dominated splitting T
U
M = E
u
⊕ E
c
with associated invariant conefield
S
u
, constants τ, n
0
satisfying (2) and (3) for every x ∈ U and n ≥ 0. Sim-
ilarly for MC. All objects except E
c
can be applied on maps in some C
2
neighbourhood of f ∈ PH(U, S
u
) to yield partial hyperbolicity. The central
distribution E
c
varies with the map, although in a continuous fashion.
The mostly contracting condition was created in [BV] to prove existence,
finitude and the no holes property of physical measures for partially hyper-
bolic diffeomorphisms. We are going to develop techniques that allow for a
generalisation of their result into a non-invertible context.
Theorem A. Every f in MC possess a finite number of physical measures
and the union of their basins of attraction cover Lebesgue almost every point
of U.
2.3 Robustness and Statistical stability
The main theorem in this paper addresses robustness properties of maps in
MC. It is not clear from the definition whether the mostly contracting con-
dition is open or not. Neither does Theorem A (nor its predecessor Theorem
A in [BV]) give any hint as to what might happen with the physical measures
under small perturbations of the map in question. In his article [D], Dolgo-
pyat addresses these kind of questions for some mostly contracting systems
on three dimensional manifolds, satisfying some additional properties which
in particular imply uniqueness of the physical measure. He achieves statis-
tical stability and strong statistical properties such as exponential decay of
correlations. The intention of this work is rather diffe rent, as we will not
bother about the number of physical measures. Nor do we study any strong
statistical properties, but will only be concerned with looking at how the
physical measures depend on the system.
10
Definition 2. Let f ∈ Diff
2
loc
(M) be a system having a finite number of phys-
ical measures µ
1
, . . . , µ
N
in some trapping region U. We say that f (strictly
speaking the pair (f, U)) is statistically stable if there exists a neighbourhood
U of f, and weakly continuous functions Φ
1
, . . . , Φ
N
: U → M(M), such
that, given any g ∈ U, the physical measures of g, supported in U, coincide
precisely with Φ
1
(g), . . . , Φ
N
(g).
Similarly, given any subset C ⊂ Diff
2
loc
(U), we define statistical stability
under perturbations within C by requiring that the functions Φ
1
, . . . , Φ
N
be
defined on C only.
Theorem B.
1. MC is open in the C
2
topology.
2. The number of physical measures supported in U is an upper semi-
continuous function MC → N.
3. Let C be any subset of MC such that the number of physical measures
supported in U is constant for maps in C. Then maps in C are statis-
tically stable under perturbations within C.
By our choice of definition, statistical stability does not make sense if
the number of physical measures changes abruptly. Theorem B states that
whenever statistical stability makes sense, it holds. In other words, a drop in
the number of physical measures is the only obstacle to statistical stability
among mostly contracting systems, so Theorem B is the strongest possible
result of its kind. Let us take a look at some of its consequences, the first of
which is immediate.
Corollary C. Maps in MC having precisely one physical measure form an
open set, and are therefore statistically stable.
As we shall see in Section 2.4, Corollary C takes a particularily nice form
when applied to conservative systems. But first, let us see how simple semi-
continuity arguments may be applied to prove great abundance of statistical
stability among mostly contracting systems.
Corollary D. Statistical stability is an open and dense property in MC.
Proof. For n ≥ 1, let S
n
be the set of maps in MC having at most n physical
measures. By semi-continuity, each S
n
is open. We define O
1
= S
1
and
O
n+1
= S
n+1
\ S
n
for every n ≥ 1. Then each O
n
is an open set on which the
number of physical measures is precisely n. Hence every map in O =
n≥1
O
n
is statistically stable and, by construction, O is dense.
11
2.4 Stable ergodicity
There is a noteworthy application of Theorem B to the theory of stable
ergodicity. We say that a diffeomorphism is conservative if it preserves
Lebesgue measure on M, and we denote the space of all conservative maps
by Diff
2
m
(M).
Definition 3. Let f ∈ Diff
2
m
(M). We say that f is stably ergodic if there
exists a C
2
neighbourhood U of f such that Lebesgue measure is ergodic under
every g ∈ U ∩ Diff
2
m
(M).
Partial hyperbolicity is b elie ved to be a strong mechanism for stable er-
godicity. See [PuSh] for details.
Corollary E. Any ergodic diffeomorphism in MC is automatically stably
ergodic.
This is not the first time stable ergodicity has been considered fror mostly
contracting systems. In [BDP], the authors give a condition (Theorem 4) of
stable ergodicity for mostly contracting systems. The point here is that
nothing at all has to be said ab out the neighbours of f, but that ergodicity
really is a robust (open) property in MC ∩ Diff
2
m
(M). Clearly the same can
be said about local diffeomorphisms, although today’s research interest in
stable ergo dicity does not reach outside the world of diffeomorphisms (as far
as I know).
2.5 Some related problems
Suppose that A, B ⊂ M are two Borel subsets, each of positive Lebesgue
measure m. To set some terminology, let us say that A and B emulsify if
supp(m
|A
) ∩ supp(m
|B
) has non-empty interior. Kan’s example [K] shows
that a mostly contracting system may possess two physical measures with
emulsifying basins.
Problem 1. Are there robust examples (in MC or elsewhere) of systems hav-
ing physical measures with basins in emulsion?
Problem 2. Do bifurcations (descontinuities in the number) of physical mea-
sures really take place for mostly contracting systems? In particular, the
example of Kan as described in [K] is an endomorphism on the cylinder
S
1
× [0, 1]. But it can easily be turned into a local diffeomorphism on the
torus T
2
by gluing two copies together. Is it then, one may ask, possible to
perform a small C
2
perturbation in such a way that the resulting system has
only one physical measure?
12
Let X be the family of all Borel subsets of M up to equivalence of zero
Lebesgue measure: A ∼ B iff m(A∆B) = 0. We endow X with the metric d
of symmetric difference on X, i.e. d(A, B) = m(A∆B) for A, B ∈ X.
Problem 3. Suppose f ∈ MC is statistically stable. Do the basins of its
physical measures vary continuously on f in the topology of symmetric dif-
ference?
2.6 Stochastic stability
We give only a brief account of noise modelling and stochastic stability of
dynamical systems, recomending [Ki] for a more detailed exposition.
Let f ∈ Diff
2
loc
(M) and {ν
}
>0
be a family of probability measures in
Diff
2
loc
(M) supported in C
2
-balls B
(f). We think of f as being a model
for a scientific phenomena and, for each , ν
is to be thought of as random
noise corresponding to external effects not accounted for by the model. The
number is the magnitude, or level of the noise.
The family {ν
}
>0
gives rise to a family {T
}
>0
of operators M(M) →
M(M) given by
T
µ =
Diff
2
loc
(M)
f
∗
µ dν
(f).
Since ν
is contained in a C
2
ball of f, it follows that
supp T
δ
x
∈ B
(f(x)) ∀x ∈ M. (4)
We refer to the property (4) by saying that the perturbations are local. In
other words, the random image of any point x is almost surely close to the
deterministic image f(x).
Another property imposed on the family {ν
}
>0
so that it provides a
realistic model of noise, is that it be absolutely continuous:
T
δ
x
<< Leb ∀x ∈ M. (5)
Being T
linear continuous, the Krylov-Bogolyubov argument proves the
existence of invariant distributions µ
= T
µ
. The set of invariant distribu-
tions is a convex subset of M(M) and, just like in the deterministic case, we
call its extreme points ergodic. Such distributions describe random orbits of
the system.
A consequence of the local property (4) is that, given a family of station-
ary dis tributions {µ
}
>0
of the corresponding T
, any weak accumulation
point µ
0
as → 0 is an f-invariant measure. Such measures are called zero
noise limits. The notion of stochastic stability is based on the idea that zero
noise limits should be compatible with physical measures.
13
Definition 4. Suppose f ∈ Diff
2
loc
(M) has some trapping region U in which
there exists a finite number of physical, say µ
1
, . . . , µ
N
. We say that f is
stochastically stable (really, the pair (f, U)), if every zero noise limit µ
0
is a
convex combination of physical measures: µ
0
= α
1
µ
1
+ . . . + α
N
µ
N
for some
non-negative α
1
, . . . , α
N
.
Traditionally, the notion of stochastic stability of an attractor assumed it
to have a unique physical measure. The definition we have given above seems
to be the natural generalisation, as no stronger property can be expected to
hold in any greater generality. See Remark D.6. in [BDV] for a discussion.
Theorem F. Every f in MC is stochastically stable.
We remark that the apparent discrepancy between statistical and stochas-
tic stability, revealed by comparing Corollary D with Theorem F, is not of a
profound nature. It merely reflects the strong definition of statistical stabil-
ity considered. Should one have settled with the weaker form of statistical
stability suggested in [V], one would obtain (quite trivially) that all mostly
contracting systems are statistically stable — not only an open and dense
set.
3 Toolbox
Whenever dealing with a normed vector space, (V, ·) say, then V (r) denotes
the ball V (r) = {v ∈ V : v < r} of radius r centred at the origin.
Given any submanifold N ⊂ M, we shall denote by d
N
(x, y) the intrinsic
distance of points x, y ∈ N defined as the infimum of arclengths of all s mooth
curves joining x and y inside N. Similarily, for x ∈ N, B
N
r
(x) denotes the
intrinsic ball {y ∈ N : d
N
(x, y) < r}.
If we are dealing with a topological space, X say, we may form the space
M(X) of Borel probability measures on X. The space M(X) is always con-
sidered with the weak topology, in which convergence µ
n
→ µ is characterised
by requiring that
ϕdµ
n
→
ϕµ for every bounded continuous ϕ : X → R.
If K ⊂ X is a subset (compact or not), we sometimes use the notation M(K)
to mean {µ ∈ M(X) : µ(K) = 1}.
3.1 Integral representation of measures
We are going to integrate measure valued functions on many occasions. The
following situation is then always understood: There are two Hausdorff spaces
X, Y and their associated spaces of Borel probability measures M(X), M(Y )
14
endowed with the weak top ology. Suppose we are given some measure µ ∈
M(X) and a continuous map ϑ : X → M(Y ). We define the measure
ϑdµ ∈ M(Y ) by requiring
ϕ d(
ϑdµ) =
ϕ dϑ(x)
dµ(x)
for every continuous ϕ : Y → R.
Alternatively, given any Borel set E ⊂ Y , we have
ϑdµ(E) =
ϑ(x)(E)dµ(x).
Measurability of the map x → ϑ(x)(E) is established by dominated pointwise
approximation of χ
E
(the indicator f unction of E) by continuous functions.
In the language of convex analysis one would say that
ϑdµ is the barycen-
tre of ϑ
∗
µ, or that ϑ
∗
µ represents
ϑdµ.
Proposition 5. The mapping µ →
ϑdµ is continuous.
Proof. Take any continuous ϕ : Y → R. Continuity of ϑ means that x →
ϕdϑ(x) is a continuous function X → R. Call it ˜ϕ. Then
ϕ d
ϑdµ =
˜ϕdµ by definition, so
ϑdµ depends indeed continuously on µ.
3.2 Admissible measures and carriers
This section introduces the notion of admissible measures, the most impor-
tant tool in this paper, used in the proof of all theorems. They should be
thought of as non-invertible analogues of Gibbs-u states (see [PeSi] for defini-
tions). Due to the non-invertibility of local diffeomorphisms, systems in PH
do not have unstable foliations. Still, there is an invariant family of manifolds
tangent to the unstable cone field. Tsujii [T] defined admissible measures for
partially hyperbolic maps with a 1-dimensional unstable direction. They are
smooth measures on an invariant family of unstable curves or, more generally,
convex combinations of such. Great care has to be taken when extending his
notion to arbitrary dimension, due to the higher geometrical complexity.
3.2.1 Admissible manifolds
We follow the approach in [ABV] for defining an invariant family of manifolds
of bounded curvature.
A C
1
embedded u-dimens ional submanifold N ⊂ M is said to be tangent
to S
u
if T
x
N ⊂ S
u
x
for every x ∈ N. Further, we say that the tangent bundle
15
of N is Lipschitz continuous if N x → T
x
N ⊂ G
u
M is a Lipschitz contin-
uous section of the Grassmannian bundle (see Section 3.2.2). The Lipschitz
variation may be quantified by c onsidering the variation of T
x
N in exponen-
tial charts. More precisely, we choose some s mall δ so that, at every x ∈ M,
the exponential map exp
x
: T
x
M(δ) → M is a diffeomorphism; and denote
by
˜
N
x
the preimage of N under exp
x
. Each point y ∈ B
δ
(x) corresponds to
a point exp
−1
x
(y) in T
x
M(δ) which we denote by ˜y. In particular, ˜x is the
zero element in T
x
M.
For every ˜y ∈
˜
N
x
, there is a unique map A
x
(y) : T
x
N → E
c
x
whose graph
is parallel to T
˜y
˜
N
x
. We say that the tangent bundle of N is K-Lipschitz
continuous at x ∈ N if A
x
(y) ≤ Kd
N
(x, y) for every ˜y ∈
˜
N
x
. Furthermore,
the tangent bundle of N is K-Lipschitz if it is K-Lipschitz at every x.
Proposition 6. Let f be partially hyperbolic. There exists a neighbourhood
U of f and K
0
> 0 such that for any g in U, and any C
1
embedded disc N
tangent to S
u
with K
0
-Lipschitz tangent bundle, the tangent bundle of g
n
(N)
has Lipshitz constant smaller than K
0
for every n > n
0
.
Proof. Fix some x ∈ N and let
˜
f
n
be the map from a neighbourhood
˜
U
x
of
the origin in T
x
M to a neighbourhood
˜
U
f
n
(x)
of the origin in T
f
n
(x)
, given by
˜
f
n
= exp
−1
f
n
(x)
◦f ◦ exp
x
.
We identify T
˜
U
x
with T
x
M (and likewise T
˜
U
f
n
(x)
with T
f
n
(x)
M) by transla-
tion. Let P be the constant field in
˜
U
f
n
(x)
associating to each z ∈
˜
U
f
n
(x)
the
subspace T
f
n
(x)
f
n
(N). We pull-back P through
˜
f
n
to obtain another field Q
in
˜
U
x
. Thus
D
˜
f
n
(˜y)Q(˜y) = T
f
n
(x)
f
n
(N) ∀˜y ∈
˜
U
x
.
To each ˜y ∈
˜
U
x
is associated a unique linear map B
x
(y) : T
x
N → E
c
x
such
that Q(˜y) is the graph of B
x
(y). Since f is C
2
, there is some C
0
> 0, uniform
in some neighbourhood U of f, such that
B
x
(y) ≤ C
0
d(x, y).
Suppose the tangent bundle of N is K-Lipschitz for some K. That is,
T
y
˜
N
x
is the graph of a uniquely defined linear map A
x
(y) : T
x
N → E
c
x
,
satisfying
A
x
(y) ≤ Kd
N
(x, y).
Therefore it is also the graph of the map
˜
A
x
(y) : Q(y) → E
c
x
given by
˜
A
x
(y) = A
x
(y) − B
x
(y).
16
We wish to estimate the norm of A
f
n
(x)
(f
n
(y)), i.e. the linear map from
T
f
n
(x)
f
n
(N) to E
c
f
n
(x)
whose graph coincides with T
˜
f
n
(˜y)
˜
N
f
n
(x)
. Note that
A
f
n
(x)
(
˜
f
n
(˜y)) = D
˜
f
n
(˜y)
|E
c
x
˜
A
x
(y)(D
˜
f
n
(˜y)
|T
x
N
)
−1
so it follows from (3) that
A
f
n
(x)
(
˜
f
n
(y)) ≤ τ
n−n
0
A
x
(y) − B
x
(y)
≤ τ
n−n
0
Kd
N
(x, y) + C
0
d(x, y)
≤ τ
n−n
0
(K + C
0
)d
N
(x, y)
≤ (τ
n−n
0
)
2
(K + C
0
)d
f
n
(N)
(f
n
(x), f
n
(y)).
The proposition follows by taking K
0
> C
0
(τ
n−n
0
)
2
1−(τ
n−n
0
)
2
.
We fix a value of K
0
once and for all as in Proposition 6.
Definition 7. We say that a u-dimensional C
1
embedded manifold is ad-
missible if it is tangent to S
u
, has a K
0
-Lipschitz tangent bundle, or is the
iterate of such under f
k
, k = 1, . . . , n
0
.
By Proposition 6, the set of admissible manifolds is invariant under iter-
ates of f. Actually, there is some C
2
neighbourhood U of f such that the
set of admissible manifolds is invariant under every g ∈ U. This ’rigidity’
property will become important in the study of small perturbations of f, bot
of random and deterministic type. Let m
N
be Lebesgue measure on some
admissible manifold. One may wonder what the possible weak accumulation
points of the sequence
1
n
n−1
k=0
f
k
∗
m
N
are. This is where admissible measures
enter the scene. They are convex combinations of smooth measures on ad-
missible manifolds. However, it is not practical to work with the space of all
admissible manifolds, but only consider a very particular kind. These will be
called carriers, because their lot in life is to ‘carry’ admissible measures.
3.2.2 The Grassmannian bundle
Recall that the u-dimensional Grassmannian manifold over a vector space
V is the set G
u
(V ) of u-dimensional subspaces of V . It can be turned into
a compact smooth u(n − u)-dimensional manifold by modelling it over the
space L(R
u
, R
n−u
) of linear maps from R
u
to R
n−u
. Namely, if H ∈ G
u
(V ),
then
L(R
u
, R
n−u
) L(H, H
⊥
) A → graph(A) ∈ G
u
(V ) (6)
defines (the inverse of) a local chart of G
u
(V ) around H; here H
⊥
is any
subspace of V , complementary to H. Let G
u
M =
x∈M
G
u
(T
x
M). It may
17
be considered as a bundle over M. Indeed, let p : G
u
M → M be the
natural projection and (U
0
, ϕ
0
) some chart on M. We define a bundle chart
ϕ
0
: p
−1
(U
0
) → U
0
×G
u
(R
u
) by ϕ
0
(x, h) = (x, Dϕ
0
(x)h). The topology given
on G
u
M is then locally the product topology of U
0
× G
u
(R
u
) induced by ϕ
0
.
In this way G
u
M b ec omes a compact manifold and the unstable conefield S
u
is a closed subset. We fix a number r
0
, small enough for the exponential map
to be a diffeomorphism on r
0
-balls in T
x
M at every x ∈ M. We will impose
further conditions on the value of r
0
later on.
3.2.3 Carriers
Having understood the notion of admissible manifolds and Grassmannian
bundle, the time is now ripe for making the notion of a carrier precise.
Definition 8. A carrier is a quadruple Γ = (r, x, h, ψ), where
• r < r
0
is a positive real number (called the radius of Γ)
• x a point in the trapping region U (called the centre of Γ)
• h ⊂ S
u
x
is a u-dimensional subspace of T
x
M (called the direction of Γ)
• ψ : h(r) → E
c
x
is a C
1
map such that
1. ψ(0) = 0,
2. Dψ(0) = 0
3. exp
x
graph(ψ) is an admissible manifold.
Recall the notation introduced in the introduction of Section 3: h(r) is the
ball {v ∈ h : v < r}. Provided that the number r
0
is small, the manifold
exp
x
graph(ψ) may be thought of as an almost round u-dimensional disc of
radius r, centred at x and tangent to h at x. The jargon we will adopt is
that we identify Γ with exp
x
graph(ψ). So a carrier Γ is in fact to be thought
of as a special kind of admissible manifold, the quadruple (r, x, h, ψ) being
its coordinates. It should be clear that if r
0
is sufficiently small, so that the
carriers are flat enough, there is only one possible centre and, consequently,
only one coordinate description of a given ‘carrier-manifold’.
The space of all carriers will be denoted by K and divided into strata
K(a), consisting of carriers with radius a. It will be given a topology in
section 3.2.6, turning it into a separable metrizable space with each stratum
K(a) being a compact subset.
18
3.2.4 Simple admissible measures
Consider some carrier Γ = (x, r, h, ψ). Let ω denote the volume form on
M derived from the Riemannian metric. Then, letting i
Γ
: Γ → M denote
inclusion, we obtain an induced volume form ω
Γ
:= i
∗
Γ
ω on Γ. We denote by
|Γ| the total mass
Γ
ω
Γ
of Γ and write (Γ, 1) for the normalised volume on
Γ:
(Γ, 1)(E) =
1
|Γ|
E∩Γ
ω
Γ
(7)
for every measurable E ⊂ M. Thus {(Γ, 1) : Γ ∈ K} is the family of
normalised Lebesgue measure on carriers. We wish to enlarge this family by
considering absolutely continuous measures with bounded densities. Suppose
φ : Γ → R is a non-negative integrable (density) function. Then we may
define the measure (Γ, φ) by
(Γ, φ)(E) =
1
|Γ|
E∩Γ
φ ω
Γ
(8)
on Borel sets E ⊂ M. The notation (Γ, 1) for the measure (7) should now
be transparent.
Definition 9. A simple admissible measure is a quintuple (r, x, h, ψ, φ) such
that Γ = (r, x, h, ψ) is a carrier and φ : Γ → R is a Borel function satisfying
•
1
|Γ|
Γ
φ ω
Γ
= 1
• log φ is bounded.
We seldom refer to a simple admissible measure explicitly as a quintuple,
but more frequently as a pair (Γ, φ). It is then understood that Γ is a carrier,
say Γ = (r, x, h, ψ), and (Γ, φ) should then be interpreted as (r, x, h, ψ, φ).
Just like a carrier, a simple admissible measure also has a radius, a centre
and a direction, given in the obvious way. By now, it should not come as a
surprise that we identify a simple admissible measure (Γ, φ) with the measure
E →
1
|Γ|
Γ∩E
φ ω
Γ
(E ⊂ M is any Borel set).
The set of all simple admissible measures is denoted by A. It can harm-
lessly be thought of as a subset of M(M). It also splits into strata A(a),
consisting of simple admissible measures of radius a. Furthermore, each
strata is the union of a nested family of sets
A(a, C) = {(r, x, h, ψ, φ) ∈ A : r = a and C
−1
≤ φ ≤ C}
19
of decreasing level of regularity. We are going to proove that, seen as a
subset of M(M), each A(a, C) is compact. The proce edure is rather prolix:
we define a topology on A using a nested fibre construction; then prove that
thus endowed, e ach A(a, C) is compact. Finally we observe that the inclusion
A(a, C) → M(M) is continuous.
3.2.5 An interlude into the heuristics of admissible measures
Let us pause for a moment to take a pee p on what is to come. It is clear that
K is not an f-invariant family. Indeed, an iterate f
n
(Γ
0
) of a carrier Γ
0
is
generally some large unshapely immersed disc that may intersect itself and
is quite far from being round. For the same reason, A cannot be invariant
under f
∗
. Still, it is quite clear that f
n
(Γ
0
) is a union of carriers, although
obviously not a disjoint one. But there is some hope that f
n
∗
(Γ
0
, 1) has an
integral representation on simple admissible measures:
f
n
∗
(Γ
0
, 1) =
A
(Γ, φ) dµ(Γ, φ) (9)
where µ is some measure on A. And so it is indeed. But since f
n
(Γ) is not a
disjoint union of carriers, the measure µ cannot be atomic. This corresponds
to the fact that one cannot cut a large disc (of dimension at least 2) into a
number of smaller ones. (The remaining objects would not look like round
discs, but bear more resemblance to half moons or, even worse, splinters of
broken porcelain.) The measure µ, at least in the way we will construct it
in sec tion 3.2.8, is not s upported on a single strata A(a). However, provided
that n is large, µ will give weight nearly 1 to some specified strata A(a). This
allows us to prove that every accumulation point of
1
n
n−1
k=0
f
k
∗
(Γ, 1) has an
integral representation of the form (9), and with µ supported on some A(a)
— a fact of great importance for the proofs of all results in this work.
3.2.6 Topology on A and K
We use bundel constructions to topologise A and K. The idea is that, locally,
K should look like a subset of the product space
R × M × G
u
(R
n
) × C
1
b
(D
u
, R
n−u
),
C
1
b
(D
u
, R
n−u
) being the space of bounded C
1
maps from the unit u-dime nsional
disc D
u
to R
n−u
and whose derivatives are also bounded. It is considered with
the usual C
1
topology.
The topology of A is to take (locally) the form of
K ×L
2
w
(D
u
).
20
Here L
2
w
(D
u
) is the space of square integrable Borel functions : D
u
→ R en-
dowed with the weak topology, in which convergence φ
n
→ φ is characterised
by requiring that
φ
n
ψd(Γ, 1) →
φψd(Γ, 1) for eve ry ψ ∈ L
2
w
(Γ).
To carry out the construction explicitly, let I be the interval (0, r
0
) and
consider the sets
˜
K =
r∈I
x∈M
h∈T
x
M
ψ∈C
1
b
(h(r),E
c
x
)
(r, x, h, ψ),
˜
A =
Γ∈
˜
K
L
2
w
(Γ).
The difference between
˜
K and K is that for a quintuple (r, x, h, ψ) to belong
to
˜
K it does not have to satisfy items (1)-(3) in the definition of carriers. We
shall define topologies on
˜
K and
˜
A and consider K and A as subsets.
Naturally, we give
r∈I
x∈M
h∈T
x
M
(r, x, h)
the topology of I × G
u
M. Thus we write
˜
K =
(r,x,h)∈I×G
u
M
C
1
b
(h(r), E
c
x
),
and intend to consider
˜
K as a vector bundle over I × G
u
M.To define the
bundle charts, fix (r
0
, x
0
, h
0
) ∈ I × G
u
M and take some local chart (V
0
, ϕ
0
)
of M around x
0
. Let p be the canonical projection
˜
K → I × G
u
M taking
(r, x, h, ψ) into (r, x, h). Write
H
0
= Dϕ(x
0
)h
0
,
D
0
= Dϕ(x
0
)h
0
(r
0
), (10)
E
0
= Dϕ(x
0
)E
c
x
0
,
so that D
0
is a u-dime nsional disc (ellipsoid) in R
n
. Each fibre C
1
b
(h(r), E
c
x
)
can be modelled over C
1
b
(D
0
, E
0
). To this end we must define a map Ψ from
p
−1
(I × p
−1
(V
0
)) to I × p
−1
(V
0
) × C
1
(D
0
, H
0
) such that
˜
K ⊃ p
−1
(I × p
−1
(V
0
))
p
Ψ
//
I × p
−1
(V
0
) × C
1
b
(D
0
, E
0
)
π
tt
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
I × p
−1
(V
0
)
commutes. Thus Ψ(r, x, h, ψ) should take the form (r, x, h, Ψ
(r,x,h)
ψ) for some
continuous linear map Ψ
(r,x,h)
: C
1
b
(h(r), E
c
x
) → C
1
b
(D
0
, E
0
). Then we take
21
neighbourhoods of (r
0
, x
0
, h
0
, ψ
0
) in
˜
K to be just the preimages, under Ψ,
of neighbourhoods of (r
0
, x
0
, h
0
, Ψ
(r
0
,x
0
,h
0
)
ψ
0
) in the product topology of I ×
p
−1
(V
0
) × C
1
b
(D
0
, E
0
).
Given any (r, x, h) ∈ V
0
there is a unique linear map A
(x,h)
such that
Dϕ
0
(x)h = graph A
(x,h)
. Let A
(x,h)
: H
0
→ R
n
be the map v → (v, A
(x,h)
v).
We define a linear map T
(r,x,h)
: H
0
→ h by
T
(r,x,h)
v =
rv
x
0
r
0
·
Dϕ(x)
−1
A
(x,h)
v
Dϕ(x)
−1
A
(x,h)
v
x
,
mapping D
0
into h(r). As H
0
and E
0
are complementary spaces, we may
identify R
n
with the product H
0
× E
0
. Let π
E
0
be the projection to the
second coordinate. Now Ψ is defined by letting
Ψ
(r,x,h)
ψ(v) = π
E
0
Dϕ(x)(T
(r,x,h)
v, ψ(T
(r,x,h)
v))
for each ψ in C
1
b
(h(r), E
c
x
).
If (r
1
, x
1
, h
1
) is another point in I × G
u
M we pick a chart (V
1
, ψ
1
) around
x
1
and produce another bundle chart
Ψ
: p
−1
(I × p
−1
(V
1
)) → I × p
−1
(V
0
) × C
1
b
(D
1
, E
1
)
in the same way. We leave it to the reader to verify that if V
0
∩ V
1
= ∅, then
Ψ
Ψ
−1
: I × p
−1
(V
0
∩ V
1
) × C
1
b
(D
0
, E
0
) → I × p
−1
(V
0
∩ V
1
) × C
1
b
(D
1
, E
1
)
is indeed a fibre preserving homeomorphism.
Proposition 10. K(a) is compact for every a ∈ (0, r
0
).
Proof. One may observe quite generally that if π : F → B is a fibre bundle
over a compact base B and C ⊂ F a subset such that
1. C is closed,
2. C ∩ π
−1
(p) is compact f or every p ∈ B,
then C is compact. The proof of Proposition 10 follows by taking B =
{a} × G
u
M, F =
˜
K(a) and C = K(a). K(a) is closed because having K
0
-
Lipschitz tangent bundle is a closed property under C
1
convergence, and each
p
−1
((a, x, h)) ∩ K(a) is compact by the Arzel`a Ascoli theorem.
22
We proceed to put a topology on
˜
A as follows. Fix Γ
0
= (r
0
, x
0
, h
0
, ψ
0
) ∈
˜
K and let W be some small neighbourhood of Γ
0
. We write q for the canonical
projection
˜
A →
˜
K. The topology we give on
˜
A is, again, locally a product
topology, obtained by turning q :
˜
A →
˜
K into a fibre bundle. Each fiber
L
2
w
(Γ) is isomorphic, via identification h(r) v → exp
x
(v, ψ(v)), to L
2
w
(D
0
).
(Here D
0
is defined as in (10)). Thus a map Φ from q
−1
(W ) to W × L
2
w
(D
0
)
must be defined so that
˜
A ⊃ q
−1
(W )
q
Φ
//
V × L
2
w
(D
0
)
π
vv
m
m
m
m
m
m
m
m
m
m
m
m
m
m
W
commutes. This requires Φ(r, x, h, ψ, φ) to take the form (r, x, h, ψ, Φ
(r,x,h,ψ)
φ).
The natural choice here is Φ
(r,x,h,ψ)
φ(v) = φ(exp
x
(T
(r,x,h)
v, ψ(T
(r,x,h)
v))). One
readily verifies that if Φ
is difined analogously to Φ over some neighbourhood
W
of a carrier Γ
1
such that W ∩ W
= ∅, then
Φ
Φ
−1
: (W ∩ W
) × L
2
w
(Γ
0
) → (W ∩ W
) × L
2
w
(Γ
1
)
is a fibre preserving homeomorphism.
Proposition 11. Every A(a, C) is compact, a ∈ I and C > 0.
Proof. We apply the same argument as in Proposition 10. All we need to
check is that each set
A(a, C)
|Γ
= {φ ∈ L
2
w
(Γ) : C
−1
≤ φ ≤ C and
φ d(Γ, 1) = 1}
is compact. Note that A(a, C)
|Γ
is contained in the ball
B
C
2
:= {φ ∈ L
2
(Γ) : φ
2
≤ C
2
}.
Since L
2
(Γ) is a Hilbert space, it is isomorphic to its dual space (L
2
(Γ))
∗
and
the weak topology on L
2
(Γ) corresponds to the weak* topology on (L
2
(Γ))
∗
.
Hence, by the Banach-Alaoglu theorem, B
C
2
is compact. Consequently, every
sequence φ
n
∈ A(a, C)
|Γ
has a weak accumulation point, i.e.
φ
n
j
ϕd(Γ, 1) →
φϕd(Γ, 1) for some subsequence φ
n
j
and every ϕ ∈ L
2
(Γ). In particular,
lim
j→∞
χ
E
φ
n
j
d(Γ, 1)
(Γ, 1)(E)
=
E
φd(Γ, 1)
(Γ, 1)(E)
∈ [C
−1
, C]
for every Borel set E ⊂ M. Hence C
−1
≤ φ ≤ C. Taking ϕ = 1 proves that
φ d(Γ, 1) = 1, so A(a, C)
|Γ
is indeed compact.
23
3.2.7 Admissible Measures
Let ι : A → M(M) be the map that associates a quintuple to its corre-
sponding measure. It is clear that ι is a continuous injection. Therefore,
each regularity level A(a, C) of each strata A(a), a < r
0
, corresponds to a
compact set ι(A(a, C)) ⊂ M(M).
Consider the space M(A) of Borel probability measures on A, endowed
with the weak topology of measures. We define the map
ι : M(A) → M(M)
µ →
ι d µ .
That is, ι(µ) is given by the Fubini-like relation
ι(µ)(E) =
A
(Γ, φ)(E) dµ(Γ, φ) (11)
for every Borel set E ⊂ M.
Definition 12. We say that µ is a lift of µ if ι(µ) = µ. A measure µ in
M(M) is said to be admissible if it has some lift in M(A).
It is useful to think of admissible measures as being convex combinations,
in an ample s ense, of simple admissible measures.
The space of admissible measures will b e denoted by AM. Furthermore,
we write AM(a, C) for the set of admissible measure that have a lift sup-
ported in A(a, C). Thus
• AM = ι(M(A)) and
• AM(a, C) = ι(M(A(a, C)))
for every 0 < a < r
0
, C > 0. Since ι : A → M(M ) is continuous, so is ι
(Proposition 5). Therefore, admissible measures of fixed radius and bounded
regularity levels form compact spaces:
Proposition 13. AM(a, C) is compact for every 0 < a < r
0
and C > 0.
Proof. The image of any continuous map from a compact space to a Hausdorff
space is compact.
Contrary to ι, ι is not injective. This means that lifts are not unique.
An easy illustration of this fact is to consider Lebesgue measure m on the
circle. Here A is understood to be the collection of all measures equivalent to
Lebesgue restricted to some interval, and whose densities density is bounded
away from zero and infinity.
24
Example 14. We may partition the circle into any finite number of curves,
say γ
1
, . . . , γ
k
. Writing α
i
= m(γ
i
) and m
|γ
i
for normalised restrictions, we
get the representation
m =
k
i=1
α
i
m
|γ
i
.
That is, m has the lift
m
1
=
k
i=1
α
i
δ
m
|γ
i
.
The lift of m
1
thus obtained is atomic: it is a convex combination of Dirac
measures. Each term corresponds to a line segment obtained by cutting the
circle. As mentioned in section 3.2.5, this cutting business cannot be used in
higher dimeinsions as it alters the geometry of objects too much, and quite
a different philosophy must be adopted.
Example 15. Consider Lebesgue measure m on the circle, just like in Example
14. Fix some small number a > 0 and, for every x ∈ S
1
, denote by m
x
the
normalised restriction of m to the interval (x − a, x + a). The measure m can
then be expressed by the relation
m =
m
x
dm(x).
In this case we obtain the lift m
2
= ξ
∗
m, where ξ : S
1
→ A(a) is the map
taking x to m
x
.
3.2.8 A disintegration technique
The lift m
2
in the previous example is in certain ways superior to m
1
. One
reason is that it perfectly reflects m, in the sense that the distribution of
the centre of carriers is given by m itself. More importantly, lifts analogous
to m
2
can be constructed in higher dimension, whereas atomic lifts like m
1
cannot. Below we set forth a general scheme to produce non-atomic lifts to
smooth measures in a more general se tting. We will be able to conclude
Proposition 16. There exists a neighbourhood U of f in PH such that AM
is invariant under every g ∈ U.
This is a rather curious fact, since A is far from being invariant. For
better comprehension, we illustrate the technique by a toy model on the
interval (0, ∞). Once understood, the general construction is a straightfor-
ward adaption, although the underlying idea gets a bit obscured by heavy
notation.
25
Example 17. Let m denote Lebesgue measure on I := (0, ∞) and R : I →
R be a function defined by R(x) = x/2. Consider the family {m
x
}
x∈I
of
normalised Lebesgue measure on I
x
:= (x − R(x), x + R(x)). We shall find a
family of densities densities φ
x
: I
x
→ R with
I
x
φ
x
dm
x
= 1, and a weight
ρ : I → R, such that
(φ
x
m
x
) d(ρm)(x) = m.
That is done by first finding any family
˜
φ
x
: I
x
→ R satisfying
(
˜
φ
x
m
x
) dm(x) = m;
then take ρ(x) =
˜
φ
x
dm
x
and normalise φ
x
= (ρ(x))
−1
˜
φ
x
.
Let V
y
= {x ∈ I : |y − x| < R(x)} = (
2y
3
, 2y). The trick is to take
˜
φ
x
(y) =
m(I
x
)
m(V
y
)
=
3x
4y
so that each
˜
φ
x
m(I
x
)
gives the same value at y, whenever x ∈ V
y
.
We have
˜
φ
x
m
x
(E) =
I
x
˜
φ
x
(y)χ
E
(y)
m(I
x
)
dm(y)
for any Borel set E ⊂ R. (Here χ
E
denotes the indicator function of E.)
Hence, by Fubini’s Theorem,
˜
φ
x
m
x
(E) dm(x) =
I
I
x
˜
φ
x
(y)χ
E
(y)
m(I
x
)
dm(y)
dm(x)
=
I
V
y
χ
E
(y)
m(V
y
)
dm(x)
dm(y) =
χ
E
(y) dm(y) = m(E)
as required. One may check that ρ(x) ≡
3
4
log 3, so the family {φ
x
}
x∈I
is
given by φ
x
(y) =
x
y log 3
.
Now suppose that (Γ, φ) = (r, p, h, ψ, φ) is some simple admissible mea-
sure. We shall prove that, although iterates of Γ under f
n
are not carriers,
push-forwards of (Γ, φ) under f
n
∗
are admissible measures. A lift of f
n
∗
(Γ, φ)
will be given explicitly and we will see in section 5.4 that this choice of lift
has some extra good properties.
Consider some iterate f
n
(Γ) of the original carrier. Since f is a local
diffeomorphism, f
|Γ
is an immersion. However, if n is large, it may happen
that f
|Γ
is not injective. In particular, f
n
(Γ) need not be a submanifold in
26
the strict sense of the word. Nevertheless, we shall associate, to each x in Γ,
a carrier denoted by Γ
x
, such that
x∈Γ
Γ
x
= f
n
(Γ).
For this purpose we define a new metric ·, ·
Γ,n
on Γ, given by the pullback
of the Riemannian metric through f
n
: For x ∈ Γ and u, v ∈ T
x
Γ, set
u, v
Γ,n
= Df
n
(x)u, Df
n
(x)v
and let d
Γ,n
(x, y) be the distance on Γ calculated using ·, ·
Γ,n
. We define a
radius function
R
a
(x) = min{a,
1
2
d
Γ,n
(x, ∂Γ)},
where a is some small number in the interval (0, r
0
). This choice makes R
a
Lipschitz continuous with constant
1
2
.
We associate, to every x ∈ Γ, the space h
x
= Df
n
(x)T
x
Γ ∈ S
u
f
n
(x)
,
and identify T
f
n
(x)
M with h
x
× E
c
f
n
(x)
. Since R
a
(x) is much smaller than
d
Γ,n
(x, ∂Γ), there is some small connected neighbourhood W
x
of x in Γ such
that exp
−1
f
n
(x)
f
n
(W
x
) is the graph of some C
1
map ψ
x
: h
x
(R
a
(x)) → E
c
f
n
(x)
.
By Proposition 6, Γ
x
:= (R
a
(x), f
n
(x), h
x
, ψ
x
) ⊂ f
n
(Γ) is a c arrier. Clearly
f
n
(Γ) =
x∈Γ
Γ
x
.
Our goal is to find de nsities φ
x
associated to each carrier Γ
y
and ρ on Γ
such that, if ξ is the map
Γ x → (Γ
x
, φ
x
) ∈ A,
then ξ
∗
(Γ, ρ) is a lift of f
n
∗
(Γ, φ). The construction of such densities will be
made in three steps.
Step 1
A neighbourhood V
y
= {x ∈ Γ : y ∈ W
z
} is assigned to every y in Γ. Let
˜
φ
x
: W
x
→ R be the family of densities given by
˜
φ
x
(y) =
φ(y)
(Γ, 1)(V
y
)
.
We claim that, given any Borel set E ⊂ M, we have
Γ
E∩W
x
˜
φ
x
d(Γ, 1)
d(Γ, 1)(x) = (Γ, φ)(E).
27
Indeed,
Γ
W
x
˜
φ
x
(y)χ
E
(y) d(Γ, 1)(y)
d(Γ, 1)(x)
=
Γ
V
y
˜
φ
x
(y)χ
E
(y) d(Γ, 1)(x)
d(Γ, 1)(y)
=
Γ
V
y
d(Γ, 1)(x)
φ(y)χ
E
(y)
(Γ, 1)(V
y
)
d(Γ, 1)(y)
=
φ(y)χ
E
(y) d(Γ, 1)(y) = (Γ, φ)(E).
Step 2
The densities
˜
φ
x
have an inconvenient defect. They are not normalised, i.e.
we do not have
W
x
˜
φ
x
d(Γ, 1) = 1
in general. We therefore write ρ(x) =
W
x
˜
φ
x
d(Γ, 1) and c onsider the nor-
malised densities
ˆ
φ
x
=
˜
φ
x
ρ(x)
,
so that indeed
W
x
ˆ
φ
x
d(Γ, 1) = 1 for every x in Γ. Moreover,
Γ
W
x
∩E
ˆ
φ
x
d(Γ, 1)
ρ(x)d(Γ, 1)(x) = (Γ, φ)(E) (12)
for every Borel set E ⊂ M. It follows from (12) that
Γ
ρ d(Γ, 1) = 1.
Step 3
In order to complete the construction of the densities φ
x
, we must transfer
the
ˆ
φ
x
from W
x
to Γ
x
. Let
J
x
(y) =
|Γ
x
|
|Γ|
| det Df
n
|T
y
Γ
|.
That is, J
x
is the Jacobian of f
n
from W
x
to Γ
x
with respect to the measures
(Γ, 1) and (Γ
x
, 1). We define φ
x
: Γ
x
→ R by
φ
x
(f
n
(y)) =
ˆ
φ
x
(y)
J
x
(y)
28
for every y ∈ W
x
. One may readily check that
(Γ
x
, φ
x
) d(Γ, ρ)(x) = f
n
∗
(Γ, φ).
Indeed, given E ⊂ M, we calculate
Γ
(Γ
x
, φ
x
)(E) d(Γ, ρ)(x)
=
Γ
Γ
x
φ
x
χ
E
d(Γ
x
, 1)
d(Γ, ρ)(x)
=
Γ
W
x
φ
x
χ
f
−n
(E)
J
x
d(Γ
x
, 1)
d(Γ, ρ)(x)
=
Γ
W
x
ˆ
φ
x
χ
f
−n
(E)
d(Γ, 1)
d(Γ, ρ)(x)
= (Γ, φ)(f
−n
(E)) = f
n
∗
(Γ, φ)(E).
Next lemma proves that all densities φ
x
in the above construction are
bounded away from zero and infinity by uniform constants, i.e. independent
of x ∈ Γ and, more importantly, independent of the iterate n ≥ 0.
Lemma 18. There exists C > 0, independent of n, such that if φ satisfies
D
−1
≤ φ ≤ D, then each φ
x
satisfies (D
2
C)
−1
≤ φ
x
≤ D
2
C. The number C
is uniform on a C
2
-neighbourhood of f.
The proof of Lemma 18 is based on a simple estimate. We use the notation
B
Γ,n
r
(x) to denote a d
Γ,n
-ball in Γ, centred at x.
Sublemma 19. We have
B
Γ,n
R
a
(x)/2
(x) ⊂ V
x
⊂ B
Γ,n
3R
a
(x)
(x)
for every x in Γ.
Proof. We prove the first inclusion. The latter is very similar. For a point
z ∈ Γ not to be in V
y
, it must satisfy
d
Γ,n
(z, y) ≥ (1 − C(r
0
))R
a
(z)
R
a
(z) ≥ R
a
(y) −
1
2
d
Γ,n
(y, z)
for some small C(r
0
) > 0 that can be choosen arbitrarily close to zero upon
reducing r
0
. Hence d
Γ,n
(y, z) ≥ (1 − C(r
0
))
2
3
R
a
(y), so V
y
contains a ball of
d
Γ,n
-radius (1 − C(r
0
))
2
3
R
a
(x) >
1
2
R
a
(x).
29
Proof of Lemma 18. Pick some x ∈ Γ. Recall that
φ
x
(f
n
(y))
φ
x
(f
n
(z))
=
φ(y)J(z)(Γ, 1)(V
z
)
φ(z)J(y)(Γ, 1)(V
y
)
for every y, z ∈ W
x
. We shall use inf and sup as shorthand notations of
ess inf and ess sup. Thus we estimate
sup φ
x
inf φ
x
≤
sup
y∈W
x
φ(y)
inf
z∈W
x
φ(z)
sup
z∈W
x
J(z)
inf
y∈W
x
J(y)
sup
z∈W
x
(Γ, 1)(V
z
)
inf
y∈W
x
(Γ, 1)(V
y
)
.
By hypothesis D
−1
≤ φ ≤ D so that
sup
y∈W
x
φ(y)
inf
z∈W
x
φ(z)
≤ D
2
.
We also know from the theory of expanding maps that there is some C
0
such
that
J
x
(z)
J
x
(y)
≤ e
C
0
d
Γ,n
(y,z)
.
Indeed, taking C
0
=
∞
k=0
τ
k−n
0
Lip(log | det Df
|Γ
|) will do just fine, but it
is wise to exagerate the value a bit so that it is holds on a neighbourhood of
f. Therefore,
sup
z∈W
x
J(z)
inf
y∈W
x
J(y)
≤ e
C
0
3R
a
(x)
.
Finally, it follows from the curvature bounds in Proposition 6 that there
exists C
1
> 1 (that can be chosen arbitrarily close to 1 upon reducing r
0
)
such that
C
−1
1
vol(B
u
)r
u
≤ (Γ, 1)(B
Γ,n
r
(x)) ≤ C
1
vol(B
u
)r
u
whenever r < d
Γ,n
(x, ∂Γ). Here vol(B
u
) is the volume of the unit ball in
u-dimensional Euclidean space. Since W
x
is contained in a ball of d
Γ,n
-radius
slightly larger than R
a
(x), say W
x
⊂ B
Γ,n
3R
a
(x)/2
(x), it follows from Sublemma
19 and the fact that R
a
is
1
2
-Lipschitz that
sup
z∈W
x
(Γ, 1)(V
z
)
inf
y∈W
x
(Γ, 1)(V
y
)
≤
C
1
vol(B
u
)3
u
(R
a
(x) +
3
4
R
a
(x))
u
C
−1
1
vol(B
u
)2
−u
(R
a
(x) −
3
4
R
a
(x))
u
= 42
u
C
2
1
Thus, taking C = e
3aC
0
C
2
1
42
u
, we arrive at
sup φ
x
inf φ
x
≤ D
2
C.
Clearly
φ d(Γ, 1) = 1 implies that inf φ
x
≤ 1 ≤ sup φ
x
, and hence
(D
2
C)
−1
≤ φ
x
≤ D
2
C.
30
Consider the map
ξ : h(r) → A (13)
x → (f
n
(x), R
a
(x), h
x
, ψ
x
, φ
x
).
Our calculations show that ξ
∗
(Γ, ρ) is indeed a lift of f
n
∗
(Γ, φ). In other
words, the push-forward under f of any s imple admissible measure is again
admissible. Of course we may do the same things for admissible measures
in general, simply by applying the machinery on every (Γ, φ) ∈ A; thus
obtaining densities, say ξ
(Γ,φ,f
n
,a)
and ρ
(Γ,φ,f
n
,a)
. We introduce the operator
Ξ
(f
n
,a)
: M(A) → M(A)
µ →
(ξ
(Γ,φ,f
n
,a)
)
∗
(Γ, ρ
(Γ,φ,f
n
,a)
)dµ(Γ, φ).
It has the property that if ι(µ) is a lift of µ, then ι(Ξ
(f
n
,a)
µ) is a lift of f
n
∗
µ.
That is,
M(A)
Ξ
(f
n
,a)
−−−−−→ M(A)
ι
ι
M(M)
f
n
∗
−−−→ M(M)
commutes. In particular, AM is invariant under every neighbour of f, so
Propostion 16 is proved. Notice that {Ξ
(f
n
,a)
: n ≥ 0} is not a semi-group.
Indeed, we do not wis h to consider iterates Ξ
(f,a)
◦ . . . ◦ Ξ
(f,a)
, for doing that
leaves us with little control regarding the radius of the simple admissible
measures over which the lifts are distributed. On the contrary, the sequence
Ξ
(f
n
,a)
µ always satisfies lim
n→∞
Ξ
(f
n
,a)
µ(A(a)) = 1 for every µ ∈ M(A).
This is because, given any simple admissible measure (Γ, φ), the set of points
x ∈ Γ for which Γ
x
has radius a grows in (Γ, φ)-measure towards 1 as n is
increased.
We identify M(M) with corresponding linear continuous functionals on
C
0
(M, R) in the usual manner by µ(ϕ) =
ϕdµ. We may consider C
0
(M, R)
∗
with its strong (or norm) topology, namely
µ
s
:= sup
ϕ∈C
0
(M,R)
ϕ
C
0
≤1
µ(ϕ).
We may extend f
∗
to the whole of C
0
(M, R)
∗
by
f
∗
µ(ϕ) = µ(ϕ ◦ f) ∀ϕ ∈ C
0
(M, R).
31
We have
f
∗
µ = sup
ϕ∈C
0
ϕ
C
0
≤1
ϕ ◦ f dµ ≤ sup
ϕ∈C
0
ϕ
C
0
≤1
ϕ dµ = µ,
so that f
∗
≤ 1. (In fact f
∗
= 1 as the equality f
∗
µ = µ clearly
holds whenever µ is a positive measure.)
Let A
1
=
a∈I
A(a, 1).
Proposition 20. Every admissible measure is strongly approximated by mea-
sures having a lift in A
1
.
Proof. A brief outline will suffice. Consider a simple admissible measure
(Γ, φ) in A(a, C). We may approximate φ in the L
1
sense by a simple func-
tion
n
i=1
a
i
χ
A
i
. Clearly, the normalised restriction (Γ, 1)
|A
i
is strongly ap-
proximated by some admissible measure µ
i
with lift in A
1
. Thus µ
(Γ,φ)
=
i
a
i
(Γ, 1)(A
i
)µ
i
is a strong approximation of (Γ, φ). The procedure can be
done simultaneously for every simple admissible measure and hence works
for admissible measures in general.
We state a non-invertible analogue of Theorem 3 in [PeSi], proving exis-
tence of Gibbs-u States for partially hyperbolic diffeomorphisms.
Proposition 21. There is a constant C > 0, uniform in a C
2
neighbourhood
of f, such that if µ
0
is any admissible measure and µ a weak accumulation
point of
1
n
n−1
k=0
f
∗
µ
0
, then µ ∈ A M(a, C) for every 0 < a < r
0
.
Proof. Take some sequence ν
i
of admissible measures with lifts in A
1
, con-
verging strongly to µ
0
. By Lemma 18, there is some large number C such that
every weak accumulation point of
1
n
n−1
k=0
f
∗
ν
i
belongs to AM(a, C). Since
f
∗
is a strong contraction on M(M), it follows by compactness of AM(a, C)
that weak accumulation points of
1
n
n−1
k=0
f
∗
µ
0
also belong to AM(a, C).
Let AM
f
be the space of f-invariant admissible measures. Proposition
21 gives this immediate corollary:
Corollary 22. AM
f
is compact.
Proof. Proposition 21 implies that there exist a, C > 0 such that AM
f
=
AM(a, C) ∩ M
f
(M). Hence by Proposition 13, AM
f
is compact.
Example 23. Let X ⊂ U be some Borel set of positive Lebesgue measure, e.g.
the basin of a physical measure. Consider the normalised restriction m
|X
of
m to X. It is not necessarily admissible, but may be strongly approximated
32
by admissible measures, say µ
i
→ m
|X
. From Proposition 21 we know that
any accumulation point µ
∞
i
of
1
n
n−1
k=0
f
k
∗
µ
i
belongs to AM
f
. Since f
∗
acts as
a contraction on C
0
(M, R)
∗
when considered with the strong topology, it fol-
lows by compactness of AM
f
that any accumulation point of
1
n
n−1
k=0
f
k
∗
m
|X
is also in AM
f
.
We take a further look at the possible uses of Proposition 21. Let S
denote the set of pairs (f, µ) in PH × M(M) such that µ belongs to AM
f
.
Proposition 24. S is a closed subset of PH × M(M).
Proof. Consider a sequence f
i
, converging to f in PH in the C
2
topology
and let µ
i
be any sequence of probabilities such that µ
i
∈ AM
f
i
for every
i. Taking a subsequence, if necessary, we may suppose that µ
i
converge to
some µ, which is necessarily an invariant measure for f. Fix some small a
and large C. By Proposition 21, every µ
i
belongs to AM(a, C), which is
compact by Proposition 13. Hence µ ∈ AM(a, C) and we are done.
3.2.9 Ergodic admissible measures
Inspired by Lemma 3.14 in [T], we prove here that every invariant admissible
measure decomposes into ergodic admissible measures. We recall Choquet’s
theorem on integral representation in locally convex spaces.
Theorem 25 (Choquet [Ph]). Let Y be a locally convex topological vector
space and X a compact convex metrisable subset. Denote by ex X the set
of extreme points of X. Then, given any point p in X, there exists a Borel
probability µ on X such that
1. µ(ex X) = 1,
2. (p) =
(x) dµ(x) for every linear continuous : Y → R.
Choquet’s theorem is often used to prove the ergodic decomposition the-
orem. Indeed, taking Y to be C(M, R)
∗
, endowed with the weak* topology
and X = M(M), one obtains
Corollary 26. Given any f-invariant probability µ, there exists a Borel prob-
ability ˆµ on M
f
(M) such that
1. ˆµ(M
erg
f
(M)) = 1,
2. µ =
M
erg
f
(M)
ν dˆµ(ν).
33
One may check that ˆµ is unique and given by
ˆµ(E) = µ
ν∈E
B(ν)
(14)
for every Borel set E ⊂ M(M). The purpose os this section is to prove an
analogous result about invariant admissible measures.
Proposition 27. Let µ be an f -invariant admissible measure. Then there
exists a unique Borel probability ˆµ on AM
f
such that
1. ˆµ(AM
erg
f
) = 1 and
2. µ =
AM
erg
f
ν dˆµ(ν).
Again, ˆµ must be given by (14), s o it is unique. The nontrivial part of the
statement is that ˆµ thus defined satisfies ˆµ(AM
erg
f
) = 1. Proposition 27 is an
immediate corollary of Choquet’s theorem and the following characterisation
of ergodic admissible measures.
Lemma 28 (Cf. proof of Lemma 3.14 in [T]). The set of extreme points
of AM
f
is precisely the set AM
erg
f
of ergodic admissible measures.
Proof. It is clear that if µ belongs to AM
erg
f
, then it is an extreme point of
AM
f
(M). Indeed, since µ cannot be written as a nontrivial convex com-
bination of distinct measures in M
f
(M), it certainly cannot be written as
a nontrivial convex combination of distinct measures in AM
f
. Conversely,
suppose that µ in AM
f
is not ergodic, say f
−1
(E) = E and 0 < µ(E) < 1.
Choose some lift µ of µ and fix > 0 arbitrarily. Given any simple admissible
measure (Γ, φ) such that (Γ, φ)(E) > 0, we may find an admissible measure
ν
(Γ,φ)
which -approximates (Γ, φ)
|E
in the strong topology. Let ν
(Γ,φ)
be lifts
of such. Then
ν :=
ν
(Γ,φ)
d µ(Γ, φ)
is a strong -approximation of µ
|E
.
Any accumulation point ν
∞
of the sequence
1
n
n−1
k=0
f
k
∗
ν is admissible.
Furthermore, since µ
|E
is f-invariant we have that ν
∞
− µ
|E
s
≤ . As is
arbitrarily s mall, it follows that
inf
ν∈AM
f
ν − µ
|E
s
= 0
so, by compactness of AM
f
, µ
|E
is admissible.
34
3.2.10 Generic Carriers
Given any µ ∈ M(M) we say that a point x is generic for µ if it is contained
in the basin B(µ). Similarily, a carrier Γ is µ-generic if (Γ, 1)-almost every
point x ∈ Γ is generic for µ. Let µ be an admissible measure. Then by
definition, there exists µ ∈ M(A) such that
µ(B(µ)) =
(Γ, φ)(B(µ)) d µ(Γ, φ). (15)
Now suppose µ is f-invariant ergodic, so that µ(B(µ)) = 1. Then the repre-
sentation (15) implies that (Γ, φ)(B(µ)) = 1 for µ-almost every (Γ, φ) ∈ A.
In particular, µ-generic carriers exist.
3.3 Stable manifolds
We briefly review Pesin’s work [Pe] on stable manifolds, an indispensible
tool in smooth ergodic theory. He proves almost everywhere existence of
stable manifolds with respect to any hyperbolic invariant measure for C
1+α
maps (α > 1). Later it has been remarked that having stable manifolds is
a pointwise property, depending on non-uniform hyperbolicity along a given
orbit, i.e. one does not have to mention any invariant measure in order to
talk about stable manifolds. Almost everywhere existence is a consequence
of Oseledet’s Theorem [O]. Moreover, stable manifolds may be constructed
for quite general sequences of diffeomorphisms (see [BP]); in particular the
theory works fine for local diffeomorphisms. Here we state a weak form of
Pesin’s theorem which, nevertheless, is quite sufficient for our needs. We
shall write N for the set of points x in U for which λ
c
+
(x) < 0.
Theorem 29. There exists a measurable function r : N → (0, ∞) satisfying
lim
n→∞
1
n
log r(f
n
(x)) = 0, (16)
and C
1
maps Ψ
x
: E
c
x
(r(x)) → E
u
x
, such that the submanifolds
W
loc
(f; x) := exp
x
graph Ψ
x
satisfy:
1. d(f
n
(x), f
n
(y)) → 0 exponentially fast as n → ∞ for every y ∈ W
loc
(f; x),
2. T
y
W
loc
(f; x) = E
c
(y) at any y ∈ W
loc
(f; x), x ∈ N ,
3. f(W
loc
(f; x)) ⊂ W
loc
(f; f(x)).
35
3.4 Absolute continuity
We fix a function r : N → (0, ∞) as in Theorem 29 so that we obtain a
family W = {W
loc
(x) : x ∈ N } of local stable leaves. Given two carriers
Γ
1
, Γ
2
, let D(h
Γ
1
,Γ
2
) be the domain {p ∈ Γ
1
: W
loc
(p) ∩ Γ
2
= ∅} ⊂ Γ
1
. It is
understood that r is small enough so that every carrier intersect every local
stable manifold in at most one point. Thus we may define the holonomy map
h
Γ
1
,Γ
2
: D(h
Γ
1
,Γ
2
) → Γ
2
p → W
loc
(p) ∩ Γ
2
.
It is clear that
(Γ
1
, 1)(D(h
Γ
1
,Γ
2
)) → 1 as Γ
2
K
→ Γ
1
.
Since the local stable manifolds are open discs, the condition W
loc
(p)∩Γ
2
= ∅
is robust under small perturbations of Γ
2
. Consequently, the map Γ
2
→
(Γ
1
, 1)(D(h
Γ
1
,Γ
2
)) is lower semi-continuous.
Let µ be the restriction of (Γ
1
, 1) to D(h
Γ
1
,Γ
2
) and ν the restriction of
(Γ
2
, 1) to h
Γ
1
,Γ
2
(D(Γ
1
, Γ
2
)) (but not normalised). We define the Jacobian of
h
Γ
1
,Γ
2
by the Radon-Nikodym derivative
Jac(h
Γ
1
,Γ
2
) =
d(h
Γ
2
,Γ
2
)
−1
∗
ν
dµ
.
Theorem 30 (Absolute Continuity [P, BP]).
1. All holonomy maps are absolutely continuous, i.e. h
Γ
1
,Γ
2
sends zero
(Γ
1
, 1)-measure sets into zero (Γ
2
, 1)-measure sets.
2. There is a uniform constant C > 0 such that
| Jac(h
Γ
1
,Γ
2
) − 1| ≤ Cd
K
(Γ
1
, Γ
2
).
Let F be any measurable union of local stable manifolds, e.g. F = N , or
F = N ∩ B(µ) for some physical measure µ.
Corollary 31. The map
A (Γ, φ) → (Γ, φ)(F) ∈ R
is lower semi-continuous.
It is a general fact that if ϕ : X → R is a lower (upper) semi-continuous
function on some probability space X, so is M(X) µ →
ϕdµ ∈ R.
Applied to the current context this becomes:
Corollary 32. The map
M(A) µ → ι(µ)(F) ∈ R
is lower semi-continuous.
36
4 Finitude of physical measures and the no
holes property
Having developed the necessary tools, we are now ready to begin the proof
of Theorem A. Although it may appear rather different, our proof resembles
that of [BV] in spirit, s imply replacing Gibbs-u states with admissible mea-
sures. Still there is one profound difference: we do not employ the technique
of Lebesgue density points when proving the no holes property.
Proof of Theorem A. Recall the statement: Every mostly contracting system
has a finite number of physical measures and the union of their basins of
attraction cover Lebesgue almost every point in the trapping region U. We
have seen in section 3.2.9 that every f in MC has some ergodic admissible
measure. The proof has three phases:
1. Every ergodic admissible measure is also a physical measure.
2. There are finitely many ergodic measures, say AM
erg
f
= {µ
1
, . . . , µ
N
}.
3. The combined basin B(µ
1
) ∪ . . . ∪ B(µ
N
) has full Lebesgue measure in
U. In particular, there is no room for yet another physical measure.
Let µ be any ergodic admissible measure and µ a lift of µ. Recall
that we denote by N the set of points in U whose maximum central Lya-
punov exponent are negative. The mostly contracting hypothesis implies
that (Γ, φ)(N ) > 0 for every simple admissible measure (Γ, φ). Hence
µ(N ) =
A
(Γ, φ)(N )d µ(Γ, φ) > 0.
The set N is f-invariat by definition, so it follows by ergodicity of µ that
µ(N ) =
A
(Γ, φ)(N )d µ(Γ, φ) = 1.
Thus µ-almost every simple admissible measure satisfies (Γ, φ)(N ) = 1. We
have seen in Section 3.2.10 that µ-almost every (Γ, φ) is µ-generic. In partic-
ular, there exists some carrier Γ such that (Γ, 1)(B(µ) ∩ N ) = 1. It follows
from absolute continuity that
A :=
x∈B(µ)∩N
W
loc
(f; x)
has positive Lebesgue measure. As A is a subset of B(µ), it follows that µ is
a physical measure.
37
Next we show that AM
erg
f
is finite. Since every ergodic admissible mea-
sure is a physical measure, there can be at most a countable number of them.
If there were to be infinitely many ergodic admissible measures, say µ
1
, µ
2
, . . .
then there would exist some sequence µ
n
j
of distinct physical measures con-
verging to some measure µ. By compactness of AM
f
, µ must be admissible.
Indeed, if µ
n
j
are lifts of the µ
n
j
in some M(A(a, C)) (see Proposition 21),
then any accumulation point µ of µ
n
j
is a lift of µ.
Writing α
n
= µ(B(µ
n
)), the ergodic decompositon of µ takes the form
µ =
∞
n=0
α
n
µ
n
. Now let B
n
= B(µ
n
) ∩ N for every n ≥ 1. Since N has
full measure with respect to any f-invariant probability, we have µ
n
(B
n
) = 1
and α
n
= µ(B
n
) for every n. Pick one k such that α
k
> 0. Using Corollary
32 we obtain
lim inf
j→∞
ι(µ
n
j
)(B
k
) ≥ ι(µ)(B
k
) = µ(B
k
) = α
k
> 0.
But this is absurd since µ
n
j
(B
k
) = 0 unless n
j
= k, which can certainly be
true for at most one value of j.
Let AM
erg
f
= {µ
1
, . . . , µ
N
}. To complete the proof of Theorem A it
remains to prove that these are the only physical measures supported in the
trapping region U, and that their combined basin B(µ
1
)∪. . .∪B(µ
N
) has full
Lebesgue measure in U. But of course the former follows from the latter. Fix
therefore some small > 0 and pick some ν
0
∈ AM with ν
0
− m
|U
s
< .
Here, m
|U
denotes the normalised restriction of Lebesgue measure to the
trapping region. Let ν
0
be any lift of ν
0
. We denote by ν
n
the averaged sums
of pus hforwards of ν
0
, and ν
n
their lifts given by the construction in section
3.2.8:
M(A) ν
0
1
n
n−1
k=0
Ξ
(f
k
,a)
−−−−−−−−−→ ν
n
∈ M(A)
ι
ι
M(M) ν
0
1
n
n−1
k=0
f
k
∗
−−−−−−→ ν
n
∈ M(M)
Let ν be an accumulation point of ν
n
. Then there is some subsequence
ν
n
j
of ν
n
, converging to a lift ν of ν. Since ν is admissible, it has an ergodic
composition of the form
ν = α
1
µ
1
+ . . . + α
N
µ
N
.
By ergodicity, µ
i
(B(µ
i
)) = 1 for each 1 ≤ i ≤ N. Hence
ν(B(µ
1
) ∪ . . . ∪ B(µ
N
)) = 1.
38
We have already seen that µ(N ) = 1 for every ergodic admissible measure.
Hence ν(N ) = 1 as well. Let F = (B(µ
1
) ∪ . . . ∪ B(µ
N
)) ∩ N . Since F is a
union of local stable manifolds, it follows from Corollary 32 that
lim inf
j→∞
ν
n
j
(F) = lim inf
j→∞
ι(ν
n
j
)(F) ≥ ι(ν)(F) = ν(F) = 1.
By invariance of F, ν
0
(F) = 1, so that
m
|U
(B(µ
1
) ∪ . . . ∪ B(µ
N
)) > 1 −
as required.
5 Robustness and statistical stability
The goal of this section is to prove Theorem B. The first part (openness
of MC) is obtained through a characterisation of the mostly contracting
hypothesis in terms of negative integrated central Lyapunov exponents for
invariant admissible measures. The remaining part of Theorem B requires
some estimates on the sizes of stable manifolds, and will be dealt with sepa-
rately.
5.1 A characterisation of the mostly contracting hy-
pothesis
The definition of maximum central Lyapunov exponent given in section 2.2
is naturally modified to take arguments in the space of invariant measures.
Recall the set S = {(f, µ) ∈ PH × M(M) : µ ∈ AM
f
}.
Definition 33. The integrated maximum central Lyapunov exponent is the
map
ˆ
λ
c
+
: S → R
(f, µ) →
λ
c
+
dµ.
Proposition 34. A partially hyperbolic system f is mostly contracting along
the central direction if and only if the integrated maximum central Lyapunov
exponent is negative on any admissible invariant measure.
Proof. The ‘only if’ was implicitly dealt with in the proof of Theorem A.
Indeed, given any admissible µ, we may write
µ(N ) =
(Γ, φ)(N ) dµ(Γ, φ)
39
for some lift µ of µ. Under the mostly contracting hypothesis we have
(Γ, φ)(N ) > 0 for every (Γ, φ) ∈ A, so µ(N ) > 0. Now, N is an f-invariant
set, so if µ is ergodic, then µ(N ) = 1. If not, it decomposes into ergodic
admissible measures, so
µ(N ) =
AM
erg
f
ν(N ) dˆµ(ν) = 1.
Hence
ˆ
λ
c
+
(µ) =
λ
c
+
dµ =
N
λ
c
+
dµ < 0 as required.
To prove the converse, choose an arbitrary C
1+Lip
disc D ⊂ U, transversial
to E
c
. Given any point p ∈ D, there is some n ≥ 0 such that f
n
(D) is
tangent to S
u
at f
n
(p). Moreover, provided that n is large enough, there
is some neighbbourhood N of p such that N is an admissible manifold. In
particular, N contains some carrier Γ. By invariance of N , it suffices to show
that (Γ, 1)(N ) > 0.
Let ν
0
= δ
(Γ,1)
and for n ≥ 1 define
ν
n
=
1
n
n−1
k=0
Ξ
(f
k
,a)
ν
0
, ν
n
= ι(ν
n
).
Again, by invariance of N , it suffices to show that ν
n
(N ) > 0 for some
n ≥ 0. Choose some convergent subsequence ν
n
j
→ ν and denote ι(ν) by ν.
We have ν ∈ AM
f
so, by hypothesis,
λ
c
+
dν < 0. Applying Corollary 32
yields ι(ν
n
j
)(N ) > 0 for every large value of j, so ν
n
j
(N ) > 0 as required.
5.2 Semi-continuity of Lyapunov exponents
It is clear that when E
c
is one-dimensional,
ˆ
λ
c
+
: S → R is continuous. In
general this property may fail, due to interaction between several central
directions. Still, it does satisfy a s emi-continuity property which is well
sufficient for our needs.
Lemma 35. The integrated maximum central Lyapunov exponent λ
c
+
: S →
R is upper semi-continuous.
Proof. Fix > 0 arbitrarily and take N large so that
1
N
log Df
N
|
E
c
dµ < λ
c
+
(f, µ) + .
Choose thereafter a neighbourhood V of (f, µ) in S, small enough for
1
N
log Dg
N
|
E
c
g
dµ
g
< λ
c
+
(f, µ) +
40
to hold for any pair (g, µ
g
) ∈ V. We have
λ
c
+
(g, µ
g
) = lim sup
n→∞
=
1
n
log Dg
n
|
E
c
g
dµ
g
≤ lim
k→∞
k−1
j=0
1
N
log Dg
N
|
E
c
g
(g
jN
(x))dµ
g
(x)
≤ λ
c
+
(µ) +
which proves the lemma.
Proof of Theorem B, part 1. Using the characterisation of the mostly con-
tracting hypothesis given by proposition 34, we find that
MC = {f ∈ PH :
ˆ
λ
c
+
(µ) < 0 ∀µ ∈ AM
f
}.
Pick some f ∈ MC. By c ompactness of AM
f
(Proposition 22) and semi-
continuity of
ˆ
λ
c
+
(Lemma 35), there is a finite collection {U
i
× V
i
}
n
i=1
⊂
PH × M(M) on which
ˆ
λ
c
+
is negative, and such that
n
i=1
U
i
× V
i
⊃ AM
f
.
Let U =
n
i=1
U
i
. Since S is closed (Proposition 24), we have
S ∩ (U × M(M)) ⊂
n
i=1
U
i
× V
i
.
Hence
ˆ
λ
c
+
is negative on AM
g
for every g ∈ U.
5.3 Large stable manifolds
The proof of the semi-continuity of the number of physical measures, as a
function on MC (part 2 of Theorem B), relies on certain estimates of the
sizes of stable manifolds. The idea is to show that the basin of each ergodic
admissible measrue is, to a certain extent, foliated by rather large stable
manifolds; and that, as a consequence of this, no other ergodic admissible
measure is allowed to lie very near it, lest their basins intersect.
Theorem 29 proves the existence of an invariant family of local stable
manifolds asso ciated to points in N . However, when dealing with basins of
measures, what one really cares about are the stable sets
W (f; x) = {y : d(f
n
(y), f
n
(x)) → 0 as n → ∞}.
For if x is in the basin of some measure µ, so is the whole of W (f; x). But
we do not know very well how W (f; x) looks in general. All we know is that
if x ∈ N , then W (f; x) contains some small embedded disc W
loc
(f; x).
Let K > 0 and define L
K
(f) to be the set of points x ∈ U for which
W (f; x) contains a disc of radius K, centred at x.
41
Lemma 36. Suppose f ∈ MC. Then there are positive constants K, θ, and
a C
2
-neighbourhood U of f such that µ(L
K
(g)) ≥ θ for every g ∈ U and
µ ∈ AM
g
.
The proof of Lemma 36 is a fairly direct consequence of an auxiliary
result regarding the existence of a large set of points with uniformly hyper-
bolic behaviour. As a consequence of Lemma 35, we may fix some small
neighbourhood U of f and a number λ < 0 such that
ˆ
λ
c
+
(g, µ) < λ < 0
for every g ∈ U and µ ∈ AM
g
(M). We also fix some small enough that
λ + 4 < 0, and N large so that
1
N
log D
c
f
N
(x)dµ(x) < λ + < 0.
Let H(g) be the set of points x ∈ M such that
n−1
j=0
D
c
g
N
(g
Nj
(x)) ≤ e
nN(λ+3)
for every n ≥ 1.
Lemma 37. There exists θ > 0 such that µ(H(g)) > θ for every g ∈ U and
µ ∈ AM
g
.
The proof of Lemma 37 is a blend of Pliss ’ Lemma and Birkhoff’s Er-
godic Theorem. The former is used to achieve good hyperbolic behaviour for
many points (positive frequency) along a fixed orbit. The latter transformes
this positive frequency into positive measure. The idea comes from Ma˜n´e’s
proof of Oseledet’s theorem [M]. However simple it may be, it is quite an
astonishing argument. For, at a first glance, it is not even clear why H(g)
should be nonempty.
Lemma 38 (Pliss’ Lemma [Pl]). Let h < A be numbers and a
0
, . . . , a
k−1
some finite sequence such that min{a
0
, . . . , a
k−1
} ≥ h and
k−1
i=0
a
i
≤ kA.
For every > 0 there exist integers 0 ≤ k
1
< . . . < k
l
< k − 1, with
l ≥ k
A+−h
, such that
n
j=k
i
a
j
≤ (n − k
i
)(A + )
for every 1 ≤ i ≤ l and k
i
≤ n ≤ k − 1.
42
A concise proof of Pliss Lemma can be encountered in [ABV].
Proof of Lemma 37. Suppose, without loss of generality, that µ is ergodic.
The general case then follows from the ergodic decomposition theorem. To
simplify notation, write ζ(x) =
1
N
log D
c
g
N
(x). By ergodicity of µ, there
is some point x
0
∈ M such that
• lim
n→∞
1
n
n−1
k=0
ζ(g
k
(x
0
)) =
ζdµ,
• lim
n→∞
1
n
#{0 ≤ k ≤ n : g
k
(x
0
) ∈ H(g)} = µ(H(g, N)).
Consider the nested sequence of sets
H
m
(g) = {x ∈ M :
n−1
j=0
D
c
g
N
(g
jN
(x)) ≤ e
nN(λ+3)
∀0 < n ≤ m}.
Clearly, H
m+1
(g) ⊂ H
m
(g) and H(g) =
m≥1
H
m
(g), so if we find θ > 0 with
µ(H
m
(g)) ≥ θ for all m ≥ 1, then we also have µ(H(g)) ≥ θ.
To this end, take some large multiple of N, say kN, satisfying
1
kN
kN −1
j=0
ζ(g
j
(x
0
)) < λ + 2.
We can decompose the orbit x
0
, g(x
0
), . . . , g
kN −1
(x
0
) into N disjoint subsets,
jumping N iterates at each time: g
j
(x
0
), g
j+N
(x
0
), . . . , g
j+(k−1)N
(x
0
), j =
0, . . . N − 1. Since the average of ζ along the x
0
, g(x
0
), . . . , g
kN −1
is less that
ζ, so must be the case for at least one of the sub-orbits. In other words, there
is at least one p ∈ {0, . . . , N − 1} satisfying
1
k
k−1
j=0
ζ(g
jN +p
(x
0
)) < λ + 2.
Let h be a lower bound for ζ, say
h = inf
g∈U
inf
x∈M
1
N
log (D
c
g
N
(x))
−1
.
According to Pliss’ Lemma, there is some l ≥ kδ, where δ =
λ+4−h
, and
0 ≤ k
1
< . . . < k
l
< k − 1, such that
1
n − k
i
n
j=k
i
ζ(g
jN +p
(x)) ≤ λ + 3
43
for each 1 ≤ i ≤ l and k
i
≤ n ≤ k − 1. Clearly k
l−m
< k − 1 − m for
each m ≥ 1, so g
k
i
N+p
(x) ∈ H
m
(g) for every i < k − 1 − m. Thus every
orbit of length kN starting at x
0
has at least l − m ≥ δk − m visits to
H
m
(g). Recall that x
0
was chosen so that the frequency of visits to H
m
(g) is
equal to µ(H
m
(g)). Therefore µ(H
m
(g)) ≥
δ
N
and the proof follows by taking
θ =
δ
N
.
Inspired by [BDP, Ta], we now dig into the proof of Lemma 36. The
tactics of the proof is to find some K > 0 such that L
K
(g) ⊃ H(g) for every
g ∈ U.
Proof of Lemma 36. Let σ = e
N(λ+4)/2
. By continuity of D
c
g, there is some
small K > 0 such that for every x ∈ H(g) we have
D
c
g
N
(y)v ≤ σv (17)
whenever d(x, y) ≤ K and v ∈ E
c
y
(g). Upon possibly reducing K we may
suppose that g
|B
K
(x)
is injective at any x ∈ M and for any g in U. We
shall prove that if x ∈ H(g), then W
s
(g, x) contains a disc of radius K,
centred at x. Indeed, it follows from (16) that we can choose j such that
Kσ
j
< r(g
jN
(x)). Let D be the disc of radius Kσ
j
, centred at g
jN
(x) in
W
s
loc
(g, g
jN
(x)). We claim that D
:= (g
|B
K
(x)
)
−jN
(D) ⊂ W
s
(g, x) contains a
disc of radius K, centred at x. The proof is by contradiction.
Suppose there exists y ∈ ∂D
with d
D
(x, y) < K. Then, by (17), we have
d
g
jN
(D
)
(g
jN
(x), g
jN
(y)) < Kσ
j
. But this is absurd, since g
jN
(y) ∈ ∂D and
D has radius Kσ
j
.
Let L(g) = {(Γ, φ) ∈ A : (Γ, φ)(L
K
(g)) > θ/2}.
Corollary 39. Let U be as in Lemma 36 and g ∈ U. Suppose µ ∈ AM
g
(M)
and let µ be any lift of µ. Then µ(L(g)) > θ/2.
Proof. Pick some number θ
< θ, and let ϕ : A → R be the function
(Γ, φ) → (Γ, φ)(L(g)).
Thus we have 0 ≤ ϕ ≤ 1 and, from Lemma 36,
ϕd µ > θ. Now take
ϕ
= 1 − ϕ. Again we have 0 ≤ ϕ
≤ 1, but this time
ϕ
d µ < 1 − θ.
Applying Chebychev’s inequality we get
µ({ϕ
≥ 1 − θ
}) ≤
ϕ
d µ
1 − θ
<
1 − θ
1 − θ
.
After rearranging we obtain µ({ϕ > θ
}) >
θ−θ
1−θ
, and the result follows by
taking θ
= θ/2.
44
5.4 The natural extension and balanced lifts
As already mentioned, lifts of admissible measures are not unique. In this
section, we define the class of balanced lifts. They are lifts with special
properties that turn out to be important in the proof of part 2 and 3 of
Theorem B. To get a flavour of what it means for a lift to be balanced,
we cheat a bit and let the reader know that the atomic lifts considered in
Example 14 are not balanced, whereas that in Example 15 is.
Given a system f ∈ PH(U, S
u
) with attractor Λ =
n≥0
f
n
(U), we asso-
ciate to it the inverse limit
ˆ
Λ
f
= {x = (. . . , x
−2
, x
−1
, x
0
) ∈ Λ
Z\N
f
: f(x
i−1
) = x
i
∀i ≤ 0},
accompanied with the map
ˆ
f :
ˆ
Λ
f
→
ˆ
Λ
f
(. . . x
−2
, x
−1
, x
0
) → (. . . x
−1
, x
0
, f(x
0
)).
Thus π ◦
ˆ
f = π ◦ f, where π is the projection to the 0th coordinate.
Elements of the inverse limit are possible histories for points in Λ
f
. Due to
the domination property of f, we may assign to each such history, a unique
direction in the Grassmannian through
g :
ˆ
Λ
f
→ G
u
M
x →
i≥0
Df
i
(x)S
u
x
−i
.
Given µ ∈ M(Λ
f
) there exists a unique measure µ
in M(
ˆ
Λ
f
), invariant
under
ˆ
f. We call µ
the natural extension of µ. We need some auxiliary
notation in order to define the notion of balanced lifts. Let Π : A → G
u
M be
the projection (r, x, h, ψ, φ) → (x, h). Whenever µ and ν are two measures
on the same measurable space and B ≥ 1 is some constant, the notation
µ
B
∼ ν means that B
−1
ν ≤ µ ≤ Bν.
Definition 40. We say that a lift µ of µ ∈ AM
f
is balanced if there is
B ≥ 1 such that Π
∗
µ
B
∼ g
∗
µ
(in which case we say that µ is B-balanced).
In particular, if µ is a balanced lift of µ, we have Π
M
∗
µ
B
∼ µ, where
Π
M
: A → M is the projection (r, x, h, ψ, φ) → x. The question arises as
to whether such lifts are always to be found. Luckily, the disintegration
technique described in Section 3.2.8 provides a mechanism to produce them
for any invariant admissible measure.
45
Proposition 41. Let f be a partially hyperbolic system. There is a neigh-
bourhood U of f in PH and B > 1 such that, given any g ∈ U and every
µ ∈ AM
g
(M), there exists a B-balanced lift for µ, supported on A(a, C).
Proof. Let µ be an f-invariant admissible measure and pick some lift µ
0
of
µ. By Proposition 21, we may suppose that µ
0
is supp orted on A(a, C) for
some large C. Since µ is invariant, µ
n
= Ξ
(f
n
,a)
µ
0
is also a lift of µ for every
n ≥ 0. We will show that any accumulation point µ of µ
n
is a B-balanced
lift of µ, and that B can be chosen uniformly in a neighbourhood of f.
We define a map
Θ : A → M(G
u
M)
(Γ, φ) →
Γ
δ
T
z
Γ
d(Γ, φ)(z),
giving the distribution of simple admissible measures in the Grassmannian
bundle. For every n ≥ 0, let ˜µ
n
= (Df
n
)
∗
Θ d µ
0
. Denoting by p the
canonical projection G
u
M → M, sending (x, h) into x, we certainly have
p
∗
˜µ
n
= f
n
∗
µ = µ for every n ≥ 0. The statement that Π
∗
µ
B
∼ g
∗
µ
results
from two claims:
1. lim
n→∞
˜µ
n
= g
∗
µ
2. Π
∗
µ
n
B
∼ ˜µ
n
for some B ≥ 1 and every n ≥ 0.
To prove the first claim, pick an open set O ⊂ G
u
M arbitrarily. We need
to prove that
lim inf
n→∞
˜µ
n
(O) ≥ g
∗
µ
(O).
Let A
i
be the set of points (x, h) ∈ O such that, given any y ∈ f
−i
(x), if
(x, h) ∈ Df
i
(S
u
y
), then Df
i
(S
u
y
) ⊂ O. The A
i
form an increasing sequence of
open sets. The domination property (3) implies that Df acts as a uniform
contraction on S
u
. Consequently, given any (x, h) ∈ O, such an i exists:
i≥0
A
i
⊃ O ∩ S
u
.
Clearly the support of g
∗
µ
is contained in S
u
. Thus, given any > 0, we
may choose some large j so that g
∗
µ
(A
j
) > g
∗
µ
(O) − . By construction
of A
j
, Df
−j
(A
j
) is an open set with the special property that
Df
−j
(A
j
) ⊃
x∈p(Df
−j
(A
j
))
S
u
x
.
46
In particular, ˜µ
n
(Df
−j
A
j
) = µ(p(Df
−j
A
j
)) = g
∗
µ
(Df
−j
(A
j
)) for every
n ≥ 0. It follows from the commuting property g ◦
ˆ
f = Df ◦ g that g
∗
µ
is
Df-invariant. We can therefore estimate
lim inf
n→∞
˜µ
n
(A
j
) = lim inf
n→∞
˜µ
n
(Df
−j
A
j
)
= µ(p(Df
−j
A
j
)) = g
∗
µ
(Df
−j
(A
j
))
= g
∗
µ
(A
j
) ≥ g
∗
µ
(O) − .
As may be taken arbitrarily small, we have indeed proved that ˜µ
n
→ g
∗
µ
.
In order to prove the second claim, note that
• ˜µ
n
=
δ
Df
n
(x)T
x
Γ
d(Γ, φ)(x)
d µ
0
(Γ, φ),
• Π
∗
µ
n
=
δ
Df
n
(x)T
x
Γ
d(Γ, ρ
(Γ,φ,f
n
,a)
)(x)
d µ
0
(Γ, φ).
Consequently, the second claim follows if we can find B such that
B
−1
φ(x) ≤ ρ
(Γ,φ,f
n
,a)
(x) ≤ Bφ(x)
for every (Γ, φ) ∈ A and x ∈ Γ. We shall prove the second inequality. The
first one is analogous.
Fix n ≥ 0 and (Γ, φ) ∈ A arbitrarily. By definition,
ρ(x) = ρ
(Γ,φ,f
n
,a)
(x) =
W
x
φ(y)
(Γ, 1)(V
y
)
d(Γ, 1)(y)
≤
sup{φ(y) : y ∈ W
x
}(Γ, 1)(W
x
)
inf{(Γ, 1)(V
y
) : y ∈ W
x
}
≤ C
2
φ(x)
(Γ, 1)(W
x
)
inf{(Γ, 1)(V
y
) : y ∈ W
x
}
.
Recall from Sublemma 19 that W
x
⊂ B
Γ,n
3R
a
(x)/2
(x). Hence (Γ, 1)(W
x
) ≤
(
3
2
)
u
C
1
vol(B
u
)R
a
(x)
u
, where C
1
is the constant described in the proof of
Lemma 18. As R
a
(x) is
1
2
-Lipschitz with res pect to the metric d
Γ,n
, we
find that inf
y∈W
x
R
a
(y) ≥ R
a
(x) −
1
2
3
2
R
a
(x) =
1
4
R
a
(x) wherefore, again by
Sublemma 19,
inf
y∈W
x
(Γ, 1)(V
y
) ≥ inf
y∈W
x
B
R
a
(x)/8
(y) ≥ 8
−u
C
−1
1
vol(B
u
)R
a
(x)
u
.
The proof follows by taking B = 12
u
C
2
C
2
1
.
47
5.5 Statistical stability
Before we indulge in the proof of parts 2 and 3 of Theorem B, let us make
some important preliminary observations. First note that a carrier (x, r, h, φ)
is quite well described by its three first coordinates, provided r is sufficiently
small. Indeed, the nonlinear displacement φ is C
1
close to exp
x
|
h
in a uniform
fashion, due to the Lipschitz condition. Now let us fix a choice of metric d
G
in G
u
M. Then, provided a is small enough, there exists δ > 0 with the
following property: Let g ∈ U, where U is as in Lemma 36, and suppose that
Γ
1
is some carrier such that (Γ
1
, 1) ∈ L(g) ∩ A(a). Then, given any carrier
Γ
2
with d
G
(Π(Γ
1
), Π(Γ
2
)) < δ, we have (Γ
1
, 1)(D(Γ
1
, Γ
2
)) > 0.
For better appreciation of the proof, let us first go through it in loose
terms. If a system g is close to f, then every physical measure ν of g is close
to the finite dimensional simplex whose vertices are the physical measures of
f, say µ
1
, . . . , µ
N
. We know that there is a fairly large portion of large stable
manifolds through many ν-generic carriers (Corollary 39). These are located
near supp µ
1
∪ . . . ∪ supp µ
N
, although we do not know precisely where. A
priori they could all be cuddled up near one of the sets supp µ
i
for which ν
tends to give positive weight. The idea is to prove that this hinders any other
physical measure ν
of g to give positive weight to µ
i
. We do that by using
proposition 41 about existence of balanced lifts. It means that if ν
were
indeed to give positive weight to (a neighbourhood of) µ
i
, then it would have
generic carriers close to some ν-generic one, possessing a big portion of large
stable manifolds. But then, as remarked above, absolute continuity of the
stable foliation would force these carriers to be generic for the same measure,
which is a contradiction. Hence every physical measure of g ’occupies’ one
physical measure of f in a one-to-one manner, so the number of physical
measures of g is at most that of f. The only way it could be equal is if every
physical measure ν
i
of g occupies precisely one of the µ
i
, and gives no weight
to the others.
Proof of Theorem 2.3 part 2 and 3. Let µ
1
, . . . , µ
N
denote the physical mea-
sures of f. As remarked upon in Section 5.4, to each µ
i
, there is a unique
inverse limit µ
i
, invariant under
ˆ
f. For every 1 ≤ i ≤ N, we cover supp µ
i
by
a finite number of balls B
ij
:= B
δ/2
(x
j
), 1 ≤ j ≤ m
i
. Thus B
i
:=
m
i
j=1
B
ij
is a neighbourhood of supp µ
i
.
Choose a C
2
neighbourhood U of f satisfying the conclusions of both
Proposition 41 and Lemma 36. Moreover, U should be small enough so
that if g ∈ U and ν
1
. . . ν
N
are the physical measures of g, then for every
1 ≤ l ≤ N
,
1. there exists 1 ≤ i ≤ N such that ν
l
(B
ij
) > 0 ∀1 ≤ j ≤ m
i
,
48
2. ν
l
(
N
i=1
B
i
) > 1 −
θ
2C
.
(The map µ → µ
is linear continuous.) The constant C is large enough so
that each ν
l
has a lift s upported on A(a, C).
Our aim is to prove that N
≤ N. Choose a small enough for the remarks
in the beginning of this section to apply. By Proposition 41, there exist B-
balanced lifts ν
1
, . . . , ν
N
of ν
1
, . . . , ν
N
, all supported on A(a, C). It follows
from the second item above that, given any ν
l
there is some ball B
ij
with
ν
l
(B
ij
∩ Π(L(g)) > 0. Since ν
l
is balanced, and ν
l
-almost every carrier
is generic for ν
l
we infer the existence of a ν
l
-generic carrier Γ
l
ij
such that
Π(Γ
l
ij
) ∈ B
ij
for some i ∈ {1, . . . , N} and j ∈ {1, . . . , m
i
}.
We claim that if k = l, then it is impossible to have ν
k
(B
ij
) > 0 for every
1 ≤ j ≤ m
i
. Otherwise, there would be some ν
k
-generic carrier Γ
k
ij
with
Π(Γ
k
ij
) ∈ B
ij
. That would imply that d
G
(Γ
l
ij
, Γ
k
ij
) < δ, and since (Γ
l
ij
, 1) ∈
L(g) we conclude (Γ
l
ij
, 1)(D(h
Γ
l
ij
,Γ
k
ij
)) > 0 which is absurd.
We have shown that each ν
l
can be associated to some µ
i
in a one-to-
one manner, namely by asking that ν
l
(B
ij
∩ Π(L(g)) be positive for some
j = 1, . . . , m
i
. Consequently N
≤ N and the second part of Theorem B is
proved.
Suppose now that N = N
. Since S is closed, each ν
l
must be close to
some convex combination α
1
µ
1
+ . . . + α
N
µ
N
. Hence ν
l
must be close to
α
1
µ
1
+. . .+α
N
µ
N
. We have already seen that ν
l
(B
ij
∩Π(L(g)) would imply
α
k
= 0 for every k = i. Hence every ν
l
is near some µ
i
and the third part of
Theorem B is proved.
6 Stochastic stability
We have seen in section 3.2.9 that for a system f ∈ PH, any invariant
admissible measure is a convex combination of ergodic admissible measures.
If, furthermore, f is has mostly contracting central direction, then every
ergodic admissible measure is a physical measure. He nce, in order to prove
that maps in MC are stochastically stable, it suffices to prove that every
zero noise limit is admissible.
Proposition 42. Let f ∈ PH and suppose that {ν
}
is a local absolutely
continuous perturbation scheme. Then every zero noise limit is admissible.
Let Ω = Diff
2
loc
(M)
Z
+
and write ν
Z
+
for the Bernoulli measure on Ω.
Given f = (f
0
, f
1
, . . .) ∈ Ω, we shall write f
n
= f
n−1
◦ . . . ◦ f
1
◦ f
0
. Replacing
f
n
with f
n
in Proposition 6 (and its proof), we obtain
49
Proposition 43. The family of admissible manifolds is f
n
-invariant for ν
Z
+
-
almost every f and every n ≥ 0, provided that is small enough.
Proposition 43 allows us to mimic the construction in section 3.2.8. In-
deed, given some simple admissible measure (Γ, φ), we consider the map
ξ
(Γ,φ,f
n
,a)
: Γ → A
x → (Γ
x
, φ
x
)
defined in section 3.2.8, along with the density ρ
(Γ,φ,f
n
,a)
on Γ, such that
(ξ
(Γ,φ,f
n
,a)
)
∗
(Γ, ρ
(Γ,φ,f
n
,a)
)
is a lift of (f
n
)
∗
(Γ, φ). Similarily, if µ lifts µ, then Ξ
(f
n
,a)
µ lifts (f
n
)
∗
µ.
Recall that T
n
µ =
Ω
(f
n
)
∗
µ d ν
Z
+
. Hence we we may define a random
operator
Ξ
rand
(ν
,n,a)
: M(A) → M(A)
µ →
Ξ
(f
n
,a)
µ d ν
Z
+
.
with the delightful property that if µ lifts µ, then Ξ
rand
(ν
,n,a)
µ lifts T
n
µ. We
infer that AM is invariant under T
.
Proof of Proposition 42. Suppose µ
is an invariant distribution under T
and
let E ⊂ M be any Borel set of zero Lebesgue measure. Then, from (5), we
have
µ
(E) = T
µ
(E) =
T
δ
x
(E) d µ
(x) = 0.
Hence µ
is absolutely continuous with respect to Lebesgue, and it follows
that it can be strongly approximated by an admissible measure. Thus given
δ > 0 arbitrarily, we may pick some admissible µ with lift µ, satisfying
µ(A
1
) = 1 (recall Proposition 20), and such that µ
−µ
s
≤ δ.
We extend T
to C
0
(M, R)
∗
by requiring
T
µ(ϕ) =
Diff
2
loc
(M)
µ(ϕ ◦ f) d ν
(f) ∀ϕ ∈ C
0
(M, R).
Given any ϕ ∈ C
0
(M) with ϕ
C
0
≤ 1, we have
T
µ(ϕ) ≤
Diff
2
loc
(M)
µ
s
d ν
(f) = µ
s
,
50
so that
T
µ
s
= sup
ϕ∈C
0
(M,R)
ϕ
C
0
≤1
T
µ(ϕ) ≤ µ
s
.
In other words, T
acts as a c ontraction on C
0
(M, R)
∗
. Hence
µ
−
1
n
n−1
k=0
T
k
µ
s
=
1
n
n−1
k=0
T
k
(µ
−µ)
s
≤ δ
for every n ≥ 0. Since
1
n
n−1
k=0
T
k
µ
0
accumulates on some AM(a, C), it
follows that
inf
µ∈AM(a,C)
µ
−µ
s
≤ δ
for every δ > 0. Therefore, by compactness, AM(a, C) must contain µ
.
By now it should now be evident that every zero noise limit is admissible.
Indeed, since AM(a, C) is a compact space, it contains any accumulation
points of stationary distributions µ
∈ AM(a, C).
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