Symbolic dynamics for three-dimensional flows with positive topological entropy

Abstract

We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C 1+ε flows on closed smooth three dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C ∞ surface has at least const ×(e hT /T ) simple closed orbits of period less than T , whenever the topological entropy h is positive -and without further assumptions on the curvature. Kat82]. It extends to flows of lesser regularity, under the additional assumption that they possess a measure of maximal entropy (Theorem 8.1). The lower bound Ce ht /T is sharp in many special cases [Hub59, Mar69, PP83, Kni97], but not in the general setup of this paper. For more on this, see §8. The theorem strengthens Katok's bound lim infWe obtain Theorem 1.1 by constructing a symbolic model that is a finite-to-one extension of ϕ. The orbits of this model are easier to understand than those of the original flow. This technique, called "symbolic dynamics", can be traced back to the work of Hadamard, Morse, Artin, and Hedlund.We proceed to describe the symbolic models used in this work. Let G be a directed graph with a countable set of vertices V . We write v → w if there is an edge from v to w, and we assume throughout that for every v there are u, w s.t.Birkhoff cocycle: Suppose r : Σ → R is a function. The Birkhoff sums of r are r n := r + r • σ + · · · + r • σ n−1 (n ≥ 1). There is a unique way to extend the definition to n ≤ 0 in such a way that the cocycle identity r m+n = r n + r m • σ n holds for all m, n ∈ Z: r 0 := 0 and r n := −r |n| • σ −|n| (n < 0). Topological Markov flow: Suppose r : Σ → R + is Hölder continuous and bounded away from zero and infinity. The topological Markov flow with roof function r and base map σ : Σ → Σ is the flow σUniform Poincaré section: The Poincaré section Λ is called uniform if its roof function is bounded away from zero and infinity. If Λ is uniform, then µ Λ is finite and it can be normalized. With this normalization, for every Borel subset E ⊂ Λ and 0All the Poincaré sections considered in this paper will be uniform, and each of them will be the disjoint union of finitely many embedded smooth two dimensional discs. Let ∂Λ denote the union of the boundaries of these discs. The set ∂Λ will introduce discontinuities to the Poincaré map of Λ. Singular set: The singular set of a Poincaré section Λ is S(Λ) :=    p ∈ Λ : p does not have a relative neighborhood V ⊂ Λ \ ∂Λ s.t. V is diffeomorphic to an open disc, and f Λ : V → f Λ (V ) and f −1 Λ : V → f −1 Λ (V ) are diffeomorphisms    . Regular set: Λ := Λ \ S(Λ). SYMBOLIC DYNAMICS FOR FLOWS 5 Basic constructions. Let ϕ be a flow satisfying our standing assumptions.Canonical transverse disc:The following lemmas are standard, see the appendix for proofs.Lemma 2.1. There is a constant r s > 0 which only depends on M and ϕ s.t. for every p ∈ M and 0 < r < r s , S := S r (p) is a C ∞ embedded closed disc, | (X q , T q S)| ≥ 1 2 radians for all q ∈ S, and dist M (·, ·) ≤ dist S (·, ·) ≤ 2 d...

Symbolic dynamics for one dimensional maps with nonuniform expansion

“…When M is compact and f is a diffeomorphism, the above statement is consequence of Propositions 3.5, 4.5 and Lemmas 4.6, 4.7 of [Sar13]. When M is compact with boundary and f is a local diffeomorphism with bounded derivatives, this is [LS19,Prop. 4.3].…”

“…The methods employed in this article require some familiarity with the articles [Sar13, LS19,LM18], and a first reading might be difficulty for those not familiar with the referred literature. Unfortunately, a self-contained exposition would lead to a lengthy manuscript, thus preventing to focus on the novelty of the work.…”

Unique equilibrium states for geodesic flows in nonpositive curvature

“…To show that the equilibrium states µ obtained in Theorem C are Bernoulli, we apply a result by Ledrappier, Lima, and Sarig [22] showing that if M is any 2-dimensional manifold, ϕ : T 1 M → R is Hölder or a scalar multiple of ϕ u , and µ is a positive entropy ergodic equilibrium measure for the geodesic flow on T 1 M , then µ is Bernoulli. Although their result is stated for positive entropy measures, this assumption is only used to guarantee that the measure has a positive Lyapunov exponent, see [23,Theorem 1.3]. Since our measure µ is hyperbolic by Corollary 3.7, it follows that [22] applies.…”

“…It follows from work of Ledrappier, Lima, and Sarig [23,22] that these equilibrium states are Bernoulli, see §9. For rank 1 surfaces, this uniqueness result is optimal; any invariant measure supported on Sing is an equilibrium state for qϕ u when q ≥ 1.…”

Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds

“…It implies that all Reeb flows on (M, ξ) have positive topological entropy. It then follows from the works of Katok [24,25], and Lima and Sarig [27] that every Reeb flow on (M, ξ) has a compact invariant set on which the dynamics is conjugated to a subshift of finite type; i.e every Reeb flow on (M, ξ) contains a "horseshoe" as a subsystem. Before giving the precise statements of our results we begin by recalling some necessary notions from contact geometry and dynamical systems.…”