Table Of ContentEXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC
SYSTEMS
MARIA CARVALHO, ANA CRISTINA MOREIRA FREITAS, JORGE MILHAZES FREITAS,
MARK HOLLAND, AND MATTHEW NICOL
Abstract. We consider the extreme value theory of a hyperbolic toral automorphism
T : T2 → T2 showing that if a Hölder observation φ which is a function of a Euclidean-
type distance to a non-periodic point ζ is strictly maximized at ζ then the corresponding
5 time series {φ◦Ti} exhibits extreme value statistics corresponding to an iid sequence of
1 random variables with the same distribution function as φ and with extremal index one.
0
Ifhoweverφisstrictlymaximizedataperiodicpointq thenthecorrespondingtime-series
2
exhibitsextremevaluestatisticscorrespondingtoaniidsequenceofrandomvariableswith
n
the same distribution function as φ but with extremal index not equal to one. We give a
a
formulafortheextremalindex(whichdependsuponthemetricusedandtheperiodofq).
J
These results imply that return times are Poisson to small balls centered at non-periodic
7
2 points and compound Poisson for small balls centered at periodic points.
]
S
D
. 1. Introduction
h
t
a
Supposewehaveatime-series{X }ofreal-valuedrandomvariablesdefinedonaprobability
m i
space (Ω,P). Extreme value theory concerns the statistical behavior of the derived time
[
series of maxima M := max{X ,...,X }. There is a well-developed theory for iid random
n 1 n
2
variables [24, 26, 14] and these references also consider the case of dependent random
v
3 variables under certain assumptions. In the setting of ergodic maps T : X → X of
2 a probability space (X,µ) modeling deterministic physical phenomena we are concerned
0
with time-series of the form {φ◦Ti} where φ : X → R is an observation of some regularity,
5
0 usually Hölder. In the literature, as in this paper, it is customary to assume that φ is
.
1 a function of the distance d(x,p) to a distinguished point ζ for some metric d so that
0 φ(x) = f(d(x,ζ)) for x ∈ X. For convenience the observation φ(x) = −logd(x,ζ) is often
5
1
Date: January 28, 2015.
:
v 2000 Mathematics Subject Classification. 37A50, 60G70, 37B20, 60G10, 37C25.
i
X Key words and phrases. Extreme Value Theory, Return Time Statistics, Stationary Stochastic Pro-
cesses, Metastability.
r
a ACMF was partially supported by FCT grant SFRH/BPD/66174/2009. JMF was partially supported
by FCT grant SFRH/BPD/66040/2009. Both these grants are financially supported by the program
POPH/FSE. ACMF and JMF are supported by FCT (Portugal) project PTDC/MAT/120346/2010,
which is financed by national and European structural funds through the programs FEDER and COM-
PETE.MC,ACMFandJMFarealsosupportedbyCMUP,whichisfinancedbyFCT(Portugal)through
the programs POCTI and POSI, with national and European structural funds, under the project PEst-
C/MAT/UI0144/2013. JMF would like to thank Mike Todd for helpful comments and suggestions.
1
2 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL
used, but scaling relations translate extreme value results for one functional to another
quite easily [9, 19]. Since the function φ◦Ti(x) detects the closer Ti(x) is to ζ, there is a
naturalrelationbetweenextremevaluestatisticsforthetimeseries{φ◦Ti}andreturntime
statistics to nested balls about ζ [4, 10]. In fact, if (X,d) is a Riemannian manifold and
µ is absolutely continuous with respect to the volume measure then return time statistics
may be deduced from extreme value statistics and vice-versa, so in some sense they are
equivalent though proved with different techniques.
Due to the close relation with return time statistics one expects a difference in the extreme
value theory of observations centered at periodic points to those centered at generic points.
We refer to the early paper of Hirata [18]. What is perhaps surprising is that in many cases
ofuniformlyexpandingsmoothonedimensionalsystemsthereisastrictdichotomybetween
the behavior of observations maximized at periodic points and other non-periodic points.
Suppose φ is maximized at a point ζ ∈ X and define a sequence of constants u by the
n
requirement that lim nµ(φ > u ) = τ, where τ ∈ R. It has been shown [7, 10, 22] that
n→∞ n
for a variety of smooth one-dimensional chaotic systems this implies that lim µ(M ≤
n→∞ n
u ) = e−ϑτ where 0 < ϑ ≤ 1 is called the extremal index and roughly measures the
n
clustering of exceedances of the maxima (more precisely ϑ−1 is the expected number of
observed exceedances if one exceedance is observed). It turns out that ϑ = 1 if ζ is not
periodic and ϑ < 1 otherwise (a precise expression for ϑ may often be given as a function
of the multiplier derivative of T along the periodic orbit). For higher dimensional smooth
chaoticsystemsasimilardichotomyisexpected(thepresenceofdiscontinuitiescomplicates
matters[1])butsofarhasnotbeenestablished. Inthispaperweestablishsuchadichotomy
forhyperbolictoralautomorphisms, giveanexpressionfortheextremalindexandfindthat
it also depends upon the metric used. This has immediate implications for the return time
statistics. WehavePoissonianreturntimestatisticstonon-periodicpointsandacompound
Poissonian distribution to periodic points (the form of the compound distribution is related
to the extremal index, as well). In fact, our results are given in terms of convergence of
pointprocessesofrareeventstothestandardPoissonprocess, inthenon-periodiccase, and
to a compound Poisson process, in the periodic case, where the multiplicity distribution
is also seen to depend on the metric used. We remark that the convergence of the point
processes is actually stronger than establishing a Poissonian distributional limit for the
number of visits to shrinking target sets.
Earlier results for invertible dynamical systems include work by Denker, Gordin and
Sharova, [5], who proved that, for toral automorphisms, the number of visits to shrinking
neighbourhoods of non-periodic points converge (when properly normalised) to a Poisson
distribution and work by Dolgopyat [6], who also proved Poissonian return time statis-
tics for non-periodic points in uniformly partially hyperbolic dynamical systems and noted
possible generalizations to the case of periodic points. There have been a variety of re-
cent results on extreme value and return time statistics for invertible dynamical systems,
both uniformly and non-uniformly hyperbolic, but restricted to observations maximized at
generic points [16, 19, 3, 13, 27, 15].
EXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC SYSTEMS 3
2. The setting
Let L : R2 → R2 be a 2×2 matrix with integer entries, with determinant equal to 1 and
without eigenvalues of absolute value 1. The eigenvalues of L are irrational numbers λ and
1/λ satisfying |λ| > 1; the corresponding eigenspaces, Es and Eu, are lines with irrational
slope and the family of lines parallel to these subspaces is invariant under L. Moreover,
L contracts vectors in Es and expands in Eu. As L(Z2) = Z2, this linear map induces an
invertible smooth transformation T on the flat torus T2 = R2/Z2 through the canonical
projection. The subspaces Es and Eu project on dense curves which intersect densely and
are T invariant; that is, the torus is the homoclinic class of the (unique) fixed point of T
and a global attractor.
After finding a norm in R2 that makes L a uniform contraction in Es and a uniform
expansion in Eu, we project the Riemannian metric generated by this norm to T2 and
consider the invariant splitting at each point into one-dimensional affine spaces parallel to
the subspaces Es and Eu. This way, T becomes an Anosov diffeomorphism: the hyperbolic
splitting at each point of T2 is obtained by translating to that point the eigenspaces of
the matrix. Through each point x ∈ T2 there is a local stable manifold Ws(x) of length
(cid:15)
(cid:15) and similarly a local unstable manifold Wu(x). Besides, T is topologically mixing and
(cid:15)
its periodic points (all saddles) are dense, even though for each q ∈ N the set of periodic
points with period q is finite.
AsthedeterminantofLisequalto1,theRiemannianstructureinducesaLebesguemeasure
m on T2 which is invariant by T. The probability measure m is Bernoulli and it is also
the unique equilibrium state of the Hölder map x ∈ T2 → ϕu(x) = −log |det DT |,
Eu(x)
whose pressure is zero. This means that the metric entropy of m is given by h (T) =
m
(cid:82)
− ϕudm = log(λ).
Unless said otherwise, we assume that we are using the Euclidean metric in R2, which we
project to T2 and denote by d(·,·). More precisely, if (x,y) denote the local coordinates of
the point z ∈ T2 then
(cid:112)
d(z,0) = x2 +y2. (2.1)
¯
Given a non-empty set A and a point x, we denote by A the closure of the set A and define
the distance of x to A by dist(x,A) := inf{d(x,y) : y ∈ A}.
Remark 2.1. Let A be a set homeomorphic to a ball whose boundary is a piecewise dif-
ferentiable Jordan curve. For ε > 0, let D := {x ∈ T2 : dist(x,A¯) ≤ ε}. Then
ε
m(D \A) = ε|∂A|+o(ε), where |·| denotes the length of the curve ∂A.
ε
Recall that a continuous function ϕ : T2 → R is α-Hölder if there are constants C > 0
and α > 0 such that |ϕ(x) − ϕ(y)| ≤ Cd(x,y)α. For α ∈ ]0,1], the space of α-Hölder
continuous functions with the norm
(cid:26) (cid:27)
|ϕ(x)−ϕ(y)|
(cid:107)ϕ(cid:107) = (cid:107)ϕ(cid:107) +sup
α C0 d(x,y)α
x(cid:54)=y
4 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL
is a Banach space. And, given k ∈ N and α ∈ ]0,1[, any α-Hölder continuous function
ϕ on the torus admits a Lipschitz approximation ϕk such that (cid:107)ϕ − ϕk(cid:107)∞ = O(k−1−αα).
From the computations in [2, Section 1.26] one can deduce that the diffeomorphism T has
exponential decay of correlations with respect to α-Hölder functions, for any α ∈ ]0,1].
More precisely, there exist constants C > 0 and 0 < θ < 1 such that, for all α-Hölder maps
φ and ψ, we have
(cid:12)(cid:90) (cid:90) (cid:90) (cid:12)
(cid:12) (cid:12)
(cid:12) φ(ψ ◦Tn)dm− φdm ψdm(cid:12) ≤ Cθn(cid:107)φ(cid:107) (cid:107)ψ(cid:107) . (2.2)
α α
(cid:12) (cid:12)
And if ψ is constant on local stable leaves then
(cid:12)(cid:90) (cid:90) (cid:90) (cid:12)
(cid:12) (cid:12)
(cid:12) φ(ψ ◦Tn)dm− φdm ψdm(cid:12) ≤ Cθn(cid:107)φ(cid:107) (cid:107)ψ(cid:107) . (2.3)
α ∞
(cid:12) (cid:12)
Among C2 Anosov diffeomorphisms the subset of the ones that admit no invariant measure
absolutely continuous with respect to m is open and dense. However, it is known that
any Anosov diffeomorphism of the torus is conjugate to a linear model; see [25]. The
conjugacy H (and its inverse) is Hölder continuous [21] and carries over the measure m
to a probability measure µ = H (m) on T2 which is invariant and mixing by the original
∗
Anosov diffeomorphism. In particular, the exponential decay of correlations (2.2) is still
valid for µ, although corresponding to different values of C, θ and α. We remark that the
conditions we will analyze in the next section, to ensure the existence of an Extreme Value
Law, are also invariant under conjugacy.
The starting point of our analysis is a stationary stochastic process X ,X ,X ,... arising
0 1 2
from the system described above in the following way. We fix some point ζ ∈ T2 and let
ϕ : T2 → R∪{+∞} be given by
ϕ(z) = −log(d(z,ζ)), (2.4)
where d denotes the projection of the usual Euclidean metric in R2 to T2, as in (2.1). We
define X = ϕ and for all i ∈ N
0
X = ϕ◦Ti. (2.5)
i
SincemisaninvariantprobabilitymeasureonT2,thenthestochasticprocessX ,X ,X ,...
0 1 2
just defined is stationary.
Note that {X > u} = B (ζ), where B (ζ) denotes the ball centered at ζ of radius e−u,
0 e−u e−u
in the Euclidean metric. In other words, having an exceedance of an high threshold u, at
time j, means that, at the jth iterate, the orbit enters the ball around ζ of radius e−u.
We remark that we could use a function different from −log in the definition of ϕ. In fact,
any not too irregular function, achieving a global maximum at 0 and invertible in a vicinity
of 0, would work. See [9, Section 1.1] for conditions on this function. For definiteness, here
we will use −log which puts us on the Type 1 or Gumbel domain of attraction for the
maxima. Our goal is to show the existence of an EVL and for that purpose the analysis
does not change much from one example to another.
EXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC SYSTEMS 5
Inordertoobtainanon-degeneratelimitforthedistributionofM := max{X ,...,X },
n 0 n−1
one usually uses normalizing sequences (u ) such that the average number of ex-
n n∈N
ceedances among the first n observations is some constant τ ≥ 0, when n is large enough,
i.e., the sequence (u ) satisfies:
n n∈N
nm(X > u ) −−−→ τ ≥ 0. (2.6)
0 n
n→∞
In the case here, if we take u = 1 log(πn)− 1 logτ, then conditions (2.6) is satisfied.
n 2 2
We are now in condition of stating our first result which asserts a dichotomy regarding
the possible limit distributions for M depending on the periodicity or aperiodicity of the
n
point ζ.
Theorem 2.2. Let T : T2 → T2 be an Anosov linear diffeomorphism. Fix ζ ∈ T2 and
define the stochastic process X ,X ,X ,... as in (2.5). Let (u ) be a sequence satisfying
0 1 2 n n∈N
(2.6) for some τ ≥ 0. Then
(1) if ζ is not periodic, we have lim m(M ≤ u ) = e−τ.
n→∞ n n
(2) if ζ is periodic of prime period q, we have lim m(M ≤ u ) = e−ϑτ, where
n→∞ n n
(cid:18) (cid:19)
ϑ = 2 arcsin √|λ|q −arcsin √ 1 .
π |λ|2q+1 |λ|2q+1
In contrast to the examples of uniformly expanding systems (with no contracting direc-
tions), the metric used plays an important role in the computation of the EI. To illustrate
this, let eu,es denote unit eigenvectors associated to the eigenvalues λ and 1/λ, respec-
tively; thus {eu,es} forms a normal basis of R2. Let z = (xu,xs), where (xu,xs) stands for
the coordinates of z in this basis. Define the metric d∗ by
d∗(z,0) = max{|xu|,|xs|} (2.7)
and then project it to T2.
Theorem 2.3. Let T : T2 → T2 be an Anosov linear diffeomorphism. Fix ζ ∈ T2 and
define the stochastic process X ,X ,X ,... as in (2.5) for the observable ϕ as in (2.4) but
0 1 2
evaluated with the metric d∗ instead. Let (u ) be a sequence satisfying (2.6) for some
n n∈N
τ ≥ 0. Then
(1) if ζ is not periodic, we have lim m(M ≤ u ) = e−τ.
n→∞ n n
(2) if ζ is periodic of prime period q, we have lim m(M ≤ u ) = e−ϑτ, where
n→∞ n n
ϑ = 1−|λ|−q.
3. The approach and dependence conditions
In order to prove the existence of EVLs in a dynamical systems context, there are a couple
of conditions on the dependence structure of the stochastic process that if verified allow
us to obtain such distributional limits. These conditions are motivated by the conditions
D(u ) and D(cid:48)(u ) of Leadbetter but were adapted to the dynamical setting and further
n n
6 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL
developed both in the absence of clustering, such as in [4, 8, 19], and in the presence of
clusteringin[10]. Veryrecently, in[12], theauthorsprovidedsomemoregeneralconditions,
called Д(u ) and Д(cid:48)(u ), which subsumed the previous ones and allowed them to address
n q n
both the presence (q ≥ 1) and the absence (q = 0) of clustering. To distinguish these
conditions the authors used a Cyrillic D to denote them. We recall these conditions here.
Given a sequence (u ) of real numbers satisfying (2.6) and q ∈ N , set
n n∈N 0
A(q) := {X > u ,X ≤ u ,...,X ≤ u }.
n 0 n 1 n q n
For s,(cid:96) ∈ N and an event B, let
s+(cid:96)−1
(cid:92)
W (B) = T−i(Bc). (3.1)
s,(cid:96)
i=s
Condition (Д (u )). We say that Д(u ) holds for the sequence X ,X ,... if, for every
q n n 0 1
(cid:96),t,n ∈ N
(cid:12)(cid:12)P(cid:0)A(q) ∩W (cid:0)A(q)(cid:1)(cid:1)−P(cid:0)A(q)(cid:1)P(cid:0)W (cid:0)A(q)(cid:1)(cid:1)(cid:12)(cid:12) ≤ γ(q,n,t), (3.2)
n t,(cid:96) n n 0,(cid:96) n
where γ(q,n,t) is decreasing in t and there exists a sequence (t ) such that t = o(n)
n n∈N n
and nγ(q,n,t ) → 0 when n → ∞.
n
For some fixed q ∈ N , consider the sequence (t ) given by condition Д (u ) and let
0 n n∈N q n
(k ) be another sequence of integers such that
n n∈N
k → ∞ and k t = o(n). (3.3)
n n n
Condition (Д(cid:48)(u )). We say that Д(cid:48)(u ) holds for the sequence X ,X ,... if there exists
q n q n 0 1
a sequence (k ) satisfying (3.3) and such that
n n∈N
(cid:98)n/kn(cid:99)
lim n (cid:88) P(cid:0)A(q) ∩T−j(cid:0)A(q)(cid:1)(cid:1) = 0. (3.4)
n n
n→∞
j=1
We note that, when q = 0, condition Д(cid:48)(u ) corresponds to condition D(cid:48)(u ) from [23].
q n n
Now let
P(A(q))
n
ϑ = lim ϑ = lim . (3.5)
n→∞ n n→∞ P(Un)
From [12, Corollary 2.4], it follows that if the stochastic process X ,X ,... satisfies both
0 1
conditions Д (u ) and Д(cid:48)(u ), the limit in (3.5) exists and
q n q n
lim P(M ≤ u ) = e−ϑτ.
n n
n→∞
Hence, the strategy to prove Theorems 2.2 and 2.3 is to show that conditions Д (u ) and
0 n
Д(cid:48)(u ) hold when ζ is a non periodic point, and that conditions Д (u ) and Д(cid:48)(u ) hold
0 n q n q n
when ζ is a periodic point of prime period q. In fact, we only check these conditions for
the usual Euclidean metric as in the context of Theorem 2.2, which is technically harder,
leaving the necessary adjustments when dealing with the adapted metric d∗ for the reader.
EXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC SYSTEMS 7
4. Checking condition Д (u )
q n
LetD beasethomeomorphictoaballwhoseboundaryispiecewisesmooth, som(∂D) = 0,
and define
(cid:8) (cid:9)
H (D) = x ∈ D : Tk(Ws(x))∩∂D (cid:54)= ∅ .
k 1
Proposition 4.1. There exist constants C > 0 and 0 < τ < 1 such that, for all k,
1
m(H (D)) ≤ Cτk. (4.1)
k 1
Proof. As a consequence of the uniform contraction within local stable manifolds, there
exists C > 0 such that dist(Tn(x),Tn(y)) ≤ C /|λ|n for all y ∈ Ws(x). In particular, this
1 1 1
implies that |Tk(Ws(x))| ≤ C /|λ|k. Therefore, for every x ∈ H (D), the leaf Tk(Ws(x))
1 1 k 1
lies in an annulus of width 2/|λ|k around ∂D. By Remark 2.1 and the invariance of m, the
result follows with τ = 1/|λ|. (cid:3)
1
Lemma 4.2. Suppose Φ : M → R is a Lipschitz map and Ψ is the indicator function
Ψ := 1W0,(cid:96)(cid:16)An(q)(cid:17).
Then for all j ≥ 0
(cid:12)(cid:90) (cid:90) (cid:90) (cid:12)
(cid:16) (cid:17)
(cid:12)(cid:12) Φ(Ψ◦Tj)dm− Φdm Ψdm(cid:12)(cid:12) ≤ C (cid:107)Φ(cid:107) τ(cid:98)j/2(cid:99) +(cid:107)Φ(cid:107) θ(cid:98)j/2(cid:99) . (4.2)
(cid:12) (cid:12) ∞ 1 Lip
Proof. We choose for reference the fixed point ζ of T and its local unstable manifold
γ˜u := Wu(ζ). By the hyperbolic product structure, each local stable manifold Ws(x)
1 1
intersects γ˜u in a unique point xˆ. Therefore, for every map Ψ we define the function
Ψ(x) := Ψ(xˆ). Also, for every i ∈ N, let Ψ = Ψ◦Ti. We note that Ψ is constant along
i i
stable manifolds and that the set of points where Ψ (cid:54)= Ψ is, by definition, the set of x for
i i
which there exist r ∈ N and x ,x on the same local stable manifold as Tr(x) such that
0 1 2
x ∈ W (cid:0)A(q)(cid:1)
1 i,(cid:96) n
but
x ∈/ W (cid:0)A(q)(cid:1).
2 i,(cid:96) n
Moreover, this set is contained in ∪i+(cid:96)−1H (A(q)). If we let i ≥ (cid:98)j/2(cid:99) then by Proposi-
k=i k n
tion 4.1 we have
∞
m((cid:8)Ψ (cid:54)= Ψ (cid:9)) ≤ (cid:88) m(H (A(q))) ≤ Cτ(cid:98)j/2(cid:99).
i i k n 1
k=(cid:98)j/2(cid:99)
By (2.3), we also get
(cid:12)(cid:90) (cid:90) (cid:90) (cid:12)
(cid:12)(cid:12) Φ(Ψ ◦Tj−(cid:98)j/2(cid:99))dm− Φdm Ψ dm(cid:12)(cid:12) ≤ C(cid:107)Φ(cid:107) (cid:107)Ψ¯(cid:107) θ(cid:98)j/2(cid:99).
(cid:98)j/2(cid:99) (cid:98)j/2(cid:99) Lip ∞
(cid:12) (cid:12)
Using the identity
(cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90)
¯ ¯ ¯ ¯
φ(ψ◦T)− φ ψ = φ(ψ◦T−ψ◦T)+ φ(ψ◦T)− φ ψ+ φ ψ− φ ψ
8 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL
we obtain
(cid:12)(cid:90) (cid:90) (cid:90) (cid:12)
(cid:0) (cid:1)
(cid:12) Φ Ψ ◦Tj−(cid:98)j/2(cid:99) dm− Φdm Ψ dm(cid:12)
(cid:12) (cid:98)j/2(cid:99) (cid:98)j/2(cid:99) (cid:12)
(cid:12)(cid:90) (cid:12)
(cid:12) (cid:0) (cid:1) (cid:12)
≤ (cid:12) Φ (Ψ −Ψ )◦Tj−(cid:98)j/2(cid:99) dm(cid:12)+C(cid:107)Φ(cid:107) θ(cid:98)j/2(cid:99)
(cid:98)j/2(cid:99) (cid:98)j/2(cid:99) Lip
(cid:12) (cid:12)
(cid:12)(cid:90) (cid:90) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
+(cid:12) Φdm Ψ −Ψ (cid:12) dm
(cid:98)j/2(cid:99) (cid:98)j/2(cid:99)
(cid:12) (cid:12)
(cid:0) (cid:8) (cid:9) (cid:1)
≤ C 2(cid:107)Φ(cid:107) m Ψ (cid:54)= Ψ +(cid:107)Φ(cid:107) θ(cid:98)j/2(cid:99)
∞ (cid:98)j/2(cid:99) (cid:98)j/2(cid:99) Lip
(cid:16) (cid:17)
≤ C (cid:107)Φ(cid:107) τ(cid:98)j/2(cid:99) +(cid:107)Φ(cid:107) θ(cid:98)j/2(cid:99) . (4.3)
∞ 1 Lip
(cid:82) (cid:82)
We complete the proof by observing that Ψdm = Ψ dm, due to the T-invariance
(cid:98)j/2(cid:99)
of m and the fact that Ψ ◦Tj−(cid:98)j/2(cid:99) = Ψ = Ψ◦Tj. (cid:3)
(cid:98)j/2(cid:99) j
To prove condition Д(u ), we will approximate the characteristic function of the set A(q)
n n
by a suitable Lipschitz function. However, this Lipschitz function will decrease sharply to
zero near the boundary of the set A(q). As the estimate in Lemma 4.2 involves the Lipschitz
n
norm, we need to bound its increase as we approach ∂A(q).
n
(cid:110) (cid:111)
Let A = A(q) and D := x ∈ A(q) : dist(cid:0)x,Ac(cid:1) ≥ n−2 , where A¯c denotes the closure
n n n n n n
of the complement of the set A . Define Φ : X → R as
n n
0 if x ∈/ A
n
Φ (x) = dist(x,Acn) if x ∈ A \D . (4.4)
n dist(x,Acn)+dist(x,Dn) n n
1 if x ∈ D
n
Note that Φ is Lipschitz continuous with Lipschitz constant given by n2. Moreover, as
n
observed in Remark 2.1, we have (cid:107)Φ −1 (cid:107) ≤ C/n2 for some constant C.
n An L1(m)
It follows that
(cid:12)(cid:90) (cid:90) (cid:12)
(cid:0) (cid:1)
(cid:12) 1 Ψ ◦Tj−(cid:98)j/2(cid:99) dm−m(A(q)) Ψdm(cid:12)
(cid:12) A(nq) (cid:98)j/2(cid:99) n (cid:12)
(cid:12)(cid:90) (cid:12)
(cid:16) (cid:17) (cid:16) (cid:17)
≤ (cid:12)(cid:12) 1 −Φ Ψ dm(cid:12)(cid:12)+C (cid:107)Φ (cid:107) j2 τ(cid:98)j/4(cid:99) +(cid:107)Φ (cid:107) θ(cid:98)j/2(cid:99)
(cid:12) An(q) n (cid:98)j/2(cid:99) (cid:12) n ∞ 1 n Lip
(cid:12)(cid:90) (cid:90) (cid:12)
(cid:16) (cid:17)
(cid:12) (cid:12)
+(cid:12) 1 −Φ dm Ψ dm(cid:12), (4.5)
(cid:12) An(q) n (cid:98)j/2(cid:99) (cid:12)
and consequently
(cid:12)(cid:12)m(cid:0)A(q) ∩W (A(q))(cid:1)−m(A(q))m(cid:0)W (A(q))(cid:1)(cid:12)(cid:12) ≤ γ(n,j)
n j,(cid:96) n n 0,(cid:96) n
where
(cid:16) (cid:17)
γ(n,j) = C n−2 +n2θ(cid:98)j/2(cid:99)
1
EXTREMAL DICHOTOMY FOR UNIFORMLY HYPERBOLIC SYSTEMS 9
and
θ = max {τ ,θ}.
1 1
Thus if, for instance, j = t = (logn)5, then nγ(n,t ) → 0 as n → ∞. Note that we have
n n
considerable freedom of choice of j in order to ensure that the previous limit is zero; taking
into account the possible applications, we chose t = (logn)5.
n
5. Checking condition Д(cid:48)(u )
q n
Before checking the condition Д(cid:48)(u ), we observe that we only need to consider the sum
q n
in (3.4) up to t = (logn)5 since, by the exponential decay of correlations, the sum of the
n
remaining terms goes to 0. More precisely, we use decay of correlations to show that
(cid:98)n/kn(cid:99)
(cid:88)
lim n m(A(q) ∩T−jA(q)) = 0. (5.1)
n n
n→∞
j=log5n
Let Φ be a suitable Lipschitz approximation of 1 , defined as in 4.4. Then
n An(q)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:90) (cid:18)(cid:90) (cid:19)2(cid:12) (cid:12)(cid:90) (cid:18)(cid:90) (cid:19)2(cid:12)
(cid:12) 1 (1 ◦Tj)dm− 1 dm (cid:12) ≤ (cid:12) Φ (Φ ◦Tj)dm− Φ dm (cid:12)
(cid:12)(cid:12) A(nq) A(nq) A(nq) (cid:12)(cid:12) (cid:12)(cid:12) n n n (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:18)(cid:90) (cid:19)2 (cid:18)(cid:90) (cid:19)2(cid:12)
+(cid:12) Φ dm − 1 dm (cid:12)
(cid:12)(cid:12) n An(q) (cid:12)(cid:12)
(cid:12)(cid:90) (cid:90) (cid:12)
(cid:12) (cid:12)
+(cid:12) 1 (1 ◦Tj)dm− Φ (Φ ◦Tj)dm(cid:12).
(cid:12) An(q) An(q) n n (cid:12)
If (logn)5 ≤ j ≤ (cid:98)n/k (cid:99), then, due to the decay of correlations, the first term is upper
n
bounded by
(cid:12) (cid:12)
(cid:12)(cid:90) (cid:18)(cid:90) (cid:19)2(cid:12)
(cid:12) Φ (Φ ◦Tj)dm− Φ dm (cid:12) ≤ Cn2θj ≤ Cn−2
(cid:12) n n n (cid:12)
(cid:12) (cid:12)
if n is sufficiently large. Recalling Remark 2.1, for the second term we obtain, for n big
enough,
(cid:12) (cid:12)
(cid:12)(cid:18)(cid:90) (cid:19)2 (cid:18)(cid:90) (cid:19)2(cid:12)
(cid:12) Φ dm − 1 dm (cid:12) ≤ Cm(A(q) \D ) ≤ Cn−2.
(cid:12)(cid:12) n An(q) (cid:12)(cid:12) n n
Similarly, we estimate the third term as follows
(cid:12)(cid:90) (cid:90) (cid:12)
(cid:12) (cid:12)
(cid:12) Φ (Φ ◦Tj)dm− 1 (1 ◦Tj)dm(cid:12) ≤ 2m(A(q) \D ) ≤ Cn−2.
(cid:12) n n An(q) An(q) (cid:12) n n
Hence equation (5.1) is satisfied.
Wu
10 M. CARVALHO, A. C. M. FREITAS, J. M. FREITAS, M. HOLLAND, AND M. NICOL
Wu Wu
B A
Ws Ws
(a) Usual Euclidean metric d (b) Adapted metric d∗
Figure 1. The sets A(q) for q > 0.
n
5.1. The periodic case: checking Д(cid:48)(u ) for q > 0. Suppose ζ is a periodic point
q n
of minimal period q for the map T. If G = Tq, then ζ is a fixed point for the Anosov
diffeomorphism G, which has eigenvalues 1 and λq, with 1 < 1 < |λ|q, where 1 and λ
λq |λ|q λ
are the eigenvalues of the original map T.
Note that A(q) corresponds to the shaded region in Figure 1a, the ball represents the set
n
{X > u }, while the ellipse is the pre-image T−q({X > u }). To compute the diameter
0 n 0 n
of A(q), we just need to find the distance between the points A and B on the intersection
n
of the ball with the ellipse. If we use local coordinates x,y for the unstable and stable
local manifolds through ζ, respectively, then the ball’s equation is x2 + y2 = s2 and the
n
ellipse’s equation is |λ|2qx2 + |λ|−2qy2 = s2, where s2 ∼ τ/(πn). Then the x coordinate
n n
of the points A and B is √ sn , while the y coordinates are √|λ|qsn and its symmetric
|λ|2q+1 |λ|2q+1
counterpart, respectively. This means that the diameter of A(q) is √2|λ|qsn .
n
|λ|2q+1
Note that, under G−j, the local stable manifolds in a small ball around ζ expand uniformly
and the local unstable manifolds contract uniformly. By construction of the sets A(q) it
n
follows that G−j(A(q))∩A(q) = ∅, while the diameter of G−j(A(q)), which is measured along
n n n
the stable direction, is less than 1. Hence, G−j(A(q)) ∩ A(q) = ∅ for all j = 1,...,g(n),
n n
where
(cid:22)logn+log(|λ|2q +1)−2log(2|λ|q(τ/π)1/2)(cid:23)
g(n) = .
2q log|λ|
Since ζ is periodic of period q, for large n we have T−jA(q)∩A(q) = ∅ for j = 1,...,qg(n).
n n
Thus,
qg(n)
(cid:88)
m(A(q) ∩T−j(A(q))) = 0. (5.2)
n n
j=1
