Dani`ele Fournier-Prunaret, Laura Gardini, & Ludwig Reich, Editors
ON EVOLUTION OF SMALL SPHERES IN THE PHASE SPACE OF A
DYNAMICAL SYSTEM
∗Boris Gurevich
1and Sergei Komech
2Abstract. We study the connection between the entropy of a dynamical system and the boundary distortion rate of regions in the phase space of the system.
AMS (2000) subject classification. 54C70, 37D20, 34D08.
Keywords. Kolmogorov entropy, Lyapunov exponents, toral automorphisms, synchronized system.
R´esum´e. Nous ´etudions la connexion entre l’entropie d’un syst`eme dynamique et le taux de distortion au bord dans l’espace des phases du syst`eme.
Mots clefs. Entropie de Kolmogorov, exposants de Lyapunov, automorphismes du tore, syst`eme synchronis´e.
Introduction
LetX = (X, ρ) be a metric space. For everyA⊂Xandε >0, we denote theε-neighborhood ofAbyOε(A). IfA={x}is a single point, thenOε(A) is the ball of radiusεcentered atx, and we denote it byB(x, ε). Let τ :X →X be a continuous map and lethµ(τ) be the Kolmogorov entropy ofτ with respect to an invariant probability measureµ. The following conjecture is motivated by some remarks in [7]:
If τ,µand a function ε7→k(ε)∈N satisfy some general conditions, then
lim ε→0
1 k(ε)ln
µ(Oε(τk(ε)B(x, ε)))
µ(B(x, ε)) =hµ(τ), (1)
where the convergence holds at least in measure.
In particular, we assume that
lim
ε→0k(ε) =∞, εlim→0k(ε)/lnε= 0. (2) (One cannot look forward to reasonable results whenkandεvary independently.)
It turns out that the conjecture is true at least within two classes of dynamical systems in some sense opposite of each other: symbolic systems (the first rigorous results for them were obtained in [2]) and smooth maps. More precisely, (1) can be proved for so-called synchronized subshifts (and hence for all sofic systems) under natural restrictions onµ, and for Anosov diffeomorphisms with SRB (see, e.g., [4]) measuresµ.
∗The work is partially supported by RFBR grant 11-01-00485-a.
1 Moscow State University & Institute for Information Transmission Problems of the Russian Academy of Sciences; email:
2 Institute for Information Transmission Problems of the Russian Academy of Sciences; email:[email protected].
c
EDP Sciences, SMAI 2012
We begin with formulating a precise result for synchronized subshifts. Then we will prove the above conjecture for hyperbolic automorphisms of tori to show some basic ideas also employed in a nonlinear case.
1.
Statement of results
Let A be a finite alphabet,S a shift transformation defined on AZ, and letX be an S-invariant compact subset ofAZ. We define a metricρonX by
ρ(x, y) =θn(x,y), x, y∈X,
whereθ∈(0,1) and
n(x, y) = min{k∈Z+:x(−k)6=y(−k) or x(k)6=y(k)} ifx6=y, andn(x, x) =∞.
We recall the definition of a synchronized system (see, e.g., [1]).
Definition 1.1. Let (X, S) be a transitive subshift. A word in the alphabetAis anX-block if it is a subblock in some x∈X. The pair (X, S) is a synchronized system if there exists an X-block w (a ’magic’ word) such that ifuw andwv (and henceu,v) are X-blocks, thenuwv is also an X-block.
It is known that the family of synchronized systems contains all subshifts of finite type and, more generally, the sofic systems introduced by Weiss [6].
Theorem 1.2. Let(X, S)be a synchronized system and letµbe an S-invariant ergodic probability measure on X. Assume that the functionk:R+→Z+ satisfies (2) and that there exists a ’magic’ wordwsuch that
µ({y∈X :y1. . . y|w|=w})>0 ,
where |w| is the length of the wordw. Then for allθ∈(0,1) equation (1) holds with convergence in the sense of L1µ.
This theorem is proved in [8].
We now remind the reader of several well-known facts concerning algebraic automorphisms of tori (see, e.g., [3, 5]).
Letτbe a hyperbolic automorphism of the torusTn=Rn/Zn and letµbe the Lebesgue (Haar) measure on Tn, which is clearlyτ-invariant.
We denote the matrix that inducesτ byAτ. Ifλ1, . . . , λmaxare its eigenvalues, then
hµ(τ) = log
Y
i:|λi|>1
|λi|. (3)
Hyperbolicity ofτ implies that|λi| 6= 1 for alli.
LetHs(Hu) be the direct sum of the subspaces inRncorresponding toλiwith|λi|<1 (|λi|>1, respectively); HsandHu are invariant underAτ,Hs∩Hu={0}, andRn=Hs⊕Hu.
We will use the following simple statement: there exist constants as>0,au >0,λs<1 and λu>1 such that for ally1∈Hs,y2∈Hu andm∈N,
||Amτy1|| ≤asλms||y1||, ||A−τmy2|| ≤auλu−m||y2||, (4)
where|| · || denotes the Euclidean norm inRn.
We denote the Lebesgue measures on Hsand Hu byνsand νu, respectively. Letλ=Qi:|λi|>1|λi|. For any measurable sets Bs⊂HsandBu⊂Hu,
Theorem 1.3. If k(ε) satisfies condition (2) and ifτ and µ are as above, then for each x∈Tn equation (1)
holds.
2.
Proof of Theorem 1.3
Let
Hs(y) :=y+Hs, Hu(y) =y+Hu, y∈Rn.
Observe that the setsHs(y) andHu(y),y∈Rn, induce two partitions ofRn that are invariant under the action ofAτ. We will keep the same notationνs andνu for the Lebesgue measures onHs(y) andHu(y), respectively. For a set D ⊂ Rn we denote its ε-neighbourhood in Rn by Oε(D). If for some y ∈ Rn, D ⊂ Hs(y) or D⊂Hu(y), we denote theε-neighbourhood ofDin the intrinsic metrics ofHs(y) orHu(y) byOsε(D) orOεu(D), respectively.
LetGs,y andGu,y be measurable subsets ofHs(y) andHu(y), respectively. We say the set
Gs,y×Gu,y:={z∈Rn: z=ys+yu−y, ys∈Gs,y, yu∈Gu,y}
is a parallelogram inRn. For ally∈Rn,δ >0, we put
Pδ(y) =Osδ(y)×Ouδ(y).
Clearly, for a given y ∈Rn, there exist constants C1, C2 depending only on the angle betweenHs and Hu such that
PC1ε(y)⊂Oε(y)⊂PC2ε(y). (6)
Upper bound. From (6) we obtain
AmτOε(y)⊂AmτPC2ε(y)
=AmτOC2ε s (y)×O
C2ε u (y)
, m∈Z+. (7)
It is easy to see that there existsγ1>0 depending only on the angle betweenHsandHu such that
Oε(Amτ PC2ε(y))⊂Oγ1ε s (A
m τO
C2ε
s (y))×O γ1ε u (A
m τO
C2ε
u (y)), m∈Z+. (8) The first inequality in (4) implies that
diam(AmτOC2ε
s (y))≤2asλms C2ε.
Sinceλs<1 and limε→0k(ε) =∞, for sufficiently smallε(sufficiently largek(ε)) we have
Ak(ε)τ OC2ε
s (y)⊂O ε s(A
k(ε) τ y)
and hence
Oγ1ε s (A
k(ε)
τ O
C2ε
s (y))⊂O ε(1+γ1)
s (A
k(ε) τ y). Therefore
νs(Osγ1ε(A k(ε)
τ O
C2ε
s (y)))≤νs(Osε(1+γ1)(A k(ε)
τ y)). (9)
Now we can estimateνu
Oγ1ε
u (AmτOCu2ε(y))
from above. By (4)
d(Amτ y, ∂(AmτOC2ε
u (y)))≥a
−1 u λ
m uC2ε,
wheredis the Euclidean metric in Rn. Then the set Θ(Ak(ε)
τ y) obtained fromAk(ε)τ OCu2ε(y) by the homothety with coefficient 2 centered atAk(ε)τ ycontains, for a sufficiently smallε, Oγu1ε(A
k(ε)
τ OCu2ε(y)). It is evident that
νu(Θ(Ak(ε)τ y)) = 2 n−lν
wherel is the dimension of Hs. Now (5) implies that
νu(Oγu1ε(A k(ε)
τ O
C2ε
u (y)))≤νu(Θ(Ak(ε)τ y)) = 2
n−lλk(ε)ν
u(OuC2ε(y)). (10)
Letν be the Lebesgue measure onRn. From (8) – (10) we conclude that
ν(Oε(Ak(ε)τ PC2ε(y)))≤
≤γ0νs(Oγs1ε(A k(ε)
τ O
C2ε
s (y)))νu(Ouγ1ε(A k(ε)
τ O
C2ε
u (y))) (11)
≤γ0α1(ε(1 +γ1)) l
2n−l λk(ε)α2 (C2ε) n−l
,
whereγ0 depends only on the angle betweenHu andHs, andα1,α2 depend only onl.
Lower bound. By (6) and the definition of the parallelogramPC1ε(y), for each m∈ Z+,
Oεs(Amτ m)×AmτOC1ε
u (y)⊂O ε(Am
τP
C1ε(y))⊂Oε(Am τO
ε(y)). (12)
Letπdenote the natural projection ofRn onto
Tn. We will use the following lemma to estimate the measure of the set on the left-hand side of (12).
Lemma 2.1. If εis small enough, then the restriction ofπ to the setOε(Ak(ε)
τ PC2ε(y))is a bijection.
Proof. From the definition of the parallelogramPC2ε(y) it follows that
diam(PC2ε(y))≤2C 2ε.
Therefore
diam(Oε(Ak(ε)τ PC2ε(y)))≤2C
2εkAτkk(ε)+ 2ε. (13) By (2) k(ε) = o(ln(ε)), hence the right-hand side of (13) tends to 0 as ε→ 0. But on every set of diameter smaller than 1 the projectionπis bijective.
The lemma just proved implies that ifεis small enough, then
ν(Oε(Ak(ε)τ PC1ε(y))) =µπOε(Ak(ε)
τ P
C1ε(y)). (14)
By (5) and (12)
ν(Oε(Ak(ε)τ PC1ε(y)))≥γ
0νs(OsεA k(ε)
τ (y))νu(OuC1ε(y))λ k(ε)
≥γ0α1εl α2 (C1ε)n−lλk(ε), (15)
whereγ0, as before, depends only on the angle betweenHsandHu. Givenx∈Tn andy∈π−1x, from (6) and (14) it follows that
ν(Oε(Ak(ε)τ PC1ε(y)))≤µ
Oε(τk(ε)B(x, ε))
≤ν(Oε(Ak(ε)τ PC2ε(y))). (16)
We substitute (11) and (15) into (16) to obtain
˜
γεn λk(ε) C1n−l≤µ
Oε(τk(ε)B(x, ε))
(17)
where ˜γ=γ0α1α2.
The measure of the ballB(x, ε) with a smallεequals
µ(B(x, ε)) =ν(Oε(y)) =α3εn, (18)
whereα3=const.
By putting (11), (17), (18) together and taking (3) into account we arrive at (1).
Remark 2.2. For a nonlinear hyperbolic map of a Riemann manifold, the proof inherits these basic construc-tions. Of course, it is more delicate to estimate the measure of the projections on the stable and unstable manifolds (in general, with nonzero curvature), and we have to use some additional technics to complete the proof. Moreover, we cannot prove (1) for allxand have to content ourself with a weaker kind of convergence.
References
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