THERMODYNAMIC FORMALISM FOR NONUNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS

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THERMODYNAMIC FORMALISM FOR NONUNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS

Abstract

This thesis examines the thermodynamic formalism of nonuniformly hyperbolic dynamical systems in two cases.

In the first part, we study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure PZ(Φ) of an almost-additive sequence Φ, on a compact finvariant set Z. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures. In the special case of subshifts of finite type we state conditions for the existence and uniqueness of Gibbs measures. We compare our results for almost-additive sequences to the thermodynamic formalism for additive sequences [Rue78, Sin72, Bow70] , nonadditive sequences [Bar96] , subadditive sequences [Fal88] , and the almostadditive sequence studied by Feng and Lau [FL02, Fen04] .

Second, we study the thermodynamic formalism for discontinuous potentials. We give conditions under which the topological pressure of a discontinuous potential can be defined. A corresponding variational principle is established, no additional conditions are required. This thermodynamic formalism is applied to nonuniformly hyperbolic maps f and the corresponding potentials ϕt(x) = −tlogJac(df|Exu). Other specific examples are considered, namely countable Markov shifts [Sar99] and unimodal maps [BK98] .

Chapter 1

Basic Notions

In this chapter the basic mathematical notions are collected. We begin with a section on hyperbolicity as it applies to this thesis. Then other preliminary definitions are given.

1.1       Hyperbolicity

A comprehensive examination of hyperbolicity as it applies to this thesis can be found in [BP02] .

A dynamical system is said to be hyperbolic if at every point there exists

an invariant splitting of the tangent bundle , where

consists of those vectors that contract under iterations of the derivative of f, and consists of those that contract under iterations of the derivative of f−1. If the splitting is continuous in x and the contraction rates do not depend on x, the system is called uniformly hyperbolic, otherwise it is called nonuniformly hyperbolic.

Let f : R → R be a diffeomorphism of a compact smooth Riemannian manifold R. Given x ∈ R and v TxR, define the Lyapunov exponent of v at x by

.

If x is fixed then the function λ(x,·) can achieve only finitely many distinct values λ(1)(x) > ··· > λ(q(x))(x). The functions λ(i)(x) and q(x) are measurable and f-invariant.

Define the f-invariant set

Λ = {x ∈ R : ∃1 ≤ k(x) < s(x) : λk(x)(x) < 0 and λk(x)+1(x) > 0}.

The dynamical system f is said to have nonzero Lyapunov exponents almosteverywhere if there exists an ergodic f-invariant Borel measure ν such that ν(Λ) = 1. The measure ν is said to be a hyperbolic measure for f.

Let µ be a hyperbolic ergodic Borel f-invariant measure on R. Thus we have that the functions and q(x) = q are constant µ almost everywhere, and there exists k(= k(x)), 1 ≤ k < q such that

.

Also, for µ almost every point x ∈ R there exist stable and unstable subspaces E(s)(x),E(u)(x) ⊂ TxR such that

  1. E(s)(x) ⊕ E(u)(x) = TxR , dfxE(s)(x) = E(s)(f(x)), and dfxE(u)(x) = E(u)(f(x)),
  2. For any n ≥ 0

,

,

where 0 < γ < 1 is a constant and C1(x) > 0 is a measurable function,

  1. ∠(Es(x),Eu(x)) ≥ D2(x) > 0, where D2(x) is a measurable function and ∠ denotes the angle between the two subspaces, and
  2. D1(fn(x)) ≤ D1(x)e, D2(fn(x)) ≥ D2(x)efor any n ≥ 0, where δ > 0 is a constant which is sufficiently small compared to 1 − γ.

The regular sets are

Λl = {x ∈ R : D1(x) ≤ l and D2(x) ≥ 1/l}.

These sets Λl are nested and exhaust Λ = Sl≥1 Λl, the set of regular points.

Theorem (Multiplicative Ergodic Theorem). If f is a C1 diffeomorphism of a compact smooth Riemannian manifold R, then the set of Lyapunov regular points has full measure with respect to any f-invariant Borel probability measure on R.

For a hyperbolic measure µ, the multiplicative ergodic theorem immediately gives us that the set of Lyapunov regular points with nonzero Lyapunov exponent contains a nonuniformly hyperbolic set of full µ measure.

A diffeomorphism f : X X is called Axiom A if the set Ω(f) of nonwandering points is hyperbolic and is the closure of the periodic points. The spectral decomposition theorem gives that Ω(f) = Ω1 S···Ss, where the basic sets Ωi are pairwise disjoint closed sets with

  1. f(Ωi) = Ωi and f|Ωi is topologically transitive,
  2. i = Zi,1 S··SZi,ni, with the Zi,j pairwise disjoint closed sets with f(Zi,j) = Zi,j+1 and f|Zi,j topologically mixing.

An Axiom A diffeomorphism is topologically conjugate to a one-sided subshift of finite type.

1.2       Preliminary Information

A comprehensive examination of the concepts and definitions defined here and used in this thesis can be found in [KH95] .

1.2.1        Subshifts of Finite and Countable Type

The set of all two-sided (double-sided) sequences of n letters is denoted

Σn = {x = …x−1x0x1 : 0 ≤ xi n − 1, for all i ∈ Z},

and the set of all one-sided (single-sided) sequences of n letters is

, for all i ∈ N}.

An n × n matrix A is a transition matrix if every entry in A is 0 or 1. We assume that the matrix is nondegenerate, that is, every row and column has at least one nonzero entry. A transition matrix gives a subset of two- (or one-) sided allowable sequences. The set of all two-sided allowable sequences is

ΣA = {x = ···x1x0x1 ··· : Axixi+1 = 1, for all i Z}, and the set of one-sided allowable sequences is

Σ+A = {x = x0x1 ··· : Axixi+1 = 1, for all i N}.

A cylinder set Cn is the set of all x ∈ Σn (or Σn, ΣA, Σ+A) is the set of all allowed sequences such that the positions −n to n (or the first n) are fixed.

The shift map on Σn, Σ+n , ΣA, and Σ+A is defined as (σ(x))i = xi+1. The systems (Σn) and (Σ+n ) are called the full two- or one-sided shift, respectively. The systems (ΣA) and (Σ+A) are called two- or one-sided subshifts of finite type. When there is no confusion, we drop the + notation for the set of one-sided sequences.

If for every i,j there exists an N = N(i,j) such that (AN)ij = 1, then the matrix A is called irreducible, and the corresponding dynamical system is topologically transitive. If there exists an N such that for every i,j (AN)ij = 1, then the matrix A is called primitive, and the corresponding dynamical system is topologically mixing. If the dynamical system is topologically mixing then there exists an N such that for every n N and every a,b of the alphabet there exists a word of length n from a to b.

1.2.2    Covers and Sets Associated to Covers; Potentials; Variations

Let U be a finite open cover of a compact metric space (X,ρ), and Wm(U) the set of all m-strings U = Ui0 …Uim of members of U, and denote by W(U) = Sm≥1 Wm(U). Let m(U) = m be the length of the string U.

Define the set

X(U) = {x X : fk(x) ∈ Uik, k = 0,…,m(U) − 1}.

We say that Γ ⊂ Wm(U) covers X if X = SU∈Γ X(U). For subshifts of finite type, if m(U) = n then X(U) is a cylinder set Cn.

Set diam(U) to be the diameter of the the collection U, that is, the largest diameter of the sets U ∈ U.

Let ϕ: X → R. The variations of ϕ are

Vn(ϕ) = sup{|ϕ(x) − ϕ(y)| : x,y X(U), m(U) = n}.

1.2.3       Measures and Entropy

Let M(X,f) be the set of f-invariant ergodic Borel probability measures on X. For any subset Z X we denote by M(Z,f) ⊂ M(X,f) the set of all f – invariant Borel probability measures on Z. If Z is compact and f-invariant, then M(Z,f) 6= ∅.

For each x X and n ≥ 0 define a probability measure µx,n on X by

,

where δx is the δ-measure supported on the point x. Denote by V (x) the set of all weak-∗ limit measures of the sequence of measures {µx,n}nN. As X is compact, ∅ 6= V (x) ⊂ M(X,f). Set

L(Z) = {x Z : V (x)M(Z,f) 6= ∅}.

The set L(Z) is not empty when Z is compact and f-invariant; L(Z) may be empty if Z is either not compact or not invariant.

For dynamical system (X,f) and measure µ ∈ M(X,f) the measuretheoretic (metric) entropy of f is denoted by hµ(f). The measure-theoretic entropy of f with respect to the partition E is denoted hµ(f,E). The entropy of µ with respect to the partition E is Hµ(E), and the conditional entropy of the partition E with respect to the partition D is denoted Hµ(E|D).

A map f is called expansive if there exists an  > 0 so that for any points x,y X with for all k ∈ Z then x = y.

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