We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each χ>0, we code a set which has full measure for every χ-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional C^∞ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of Hölder continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

We present some developments in the study of chaotic dynamics following the solution of a conjecture of Newhouse on the measures maximizing the entropy of smooth surface diffeomorphisms. We focus on strong positive recurrence, a generalization of the classical Anosov-Smale theory of uniform hyperbolicity introduced in a joint work with Sylvain Crovisier and Omri Sarig. This new property is general enough to be satisfied by all smooth surface diffeomorphisms with positive entropy, yet it still ensures many quantitative properties such as exponential mixing or limit theorems for regular functions. We also present some open problems, including its abundance (or not) in higher dimensions.

Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.

We introduce the strong positive recurrence (SPR) property for diffeomorphisms on closed manifolds with arbitrary dimension, and show that it has many consequences and holds in many cases. SPR diffeomorphisms can be coded by countable state Markov shifts whose transition matrices act with a spectral gap on a large Banach space, and this implies exponential decay of correlations, almost sure invariance principle, large deviations, among other properties of the ergodic measures of maximal entropy. Any C^∞ smooth surface diffeomorphism with positive entropy is SPR, and there are many other examples with lesser regularity, or in higher dimension.

We construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each $\chi>0$, we code a set of full measure for every invariant probability measure which is $\chi$-hyperbolic. These include all ergodic measures with entropy bigger than $\chi$ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of $[-\chi,\chi]$. This contrasts with a previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts.

Let f be a C^r surface diffeomorphism with large entropy (more precisely, h_ top(f)>λ_min(f)/r). Then the number of ergodic measures of maximal entropy is upper semicontinuous at f. This generalizes the C^∞ case studied in , answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of f admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the C^∞ case.

We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending investigation of the quadratic case by Leplaideur and Watbled. Under suitable conditions, we prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures. In this non-linear thermodynamical formalism, which can, e.g., model meanfield approximation of large systems, several kind of phase transitions appear, some of which cannot happen in the linear case. We use our correspondence between non-linear and linear equilibrium measures to further the understanding of phase transitions, both in previously known cases (Curie-Weiss and Potts models) and in new examples (metastable phase transition). Finally, we apply some of the ideas introduced to the classical thermodynamical formalism, proving that freezing phase transitions can occur over any zero-entropy invariant compact subset of the phase space.

This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.

We give a proof of Viana’s [19] conjecture on physical measures, in the special case of $C^\infty $ surface diffeomorphisms, and using the analysis of entropy and Lyapunov exponents we developed in [5]. Burguet [3] has recently proved a stronger result, using a different method.

We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.

We show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.

We study the entropy and Lyapunov exponents of invariant measures μ for smooth surface diffeomorphisms f, as functions of (f,μ). The main result is an inequality relating the discontinuities of these functions. One consequence is that for a C^∞ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.

We study the one parameter family of potential functions q(phi)(u) associated with the unstable Jacobian potential (or geometric potential) q(u) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For q < 1, it is known that there is a unique equilibrium state associated with q(phi)(u), and it has full support. For q > 1 it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value q = 1 and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure or measures supported on the singular set. In particular, when q = 1, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Holder-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms. As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinai: billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a Holder-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.

A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound deriving from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). Adapting classical techniques, we use perturbations that are local and can be chosen to preserve volume or symplectic form or a homoclinic connection. This optimal entropy creation yields a number of consequences for $C^1$-generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism and we prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero.

Extending work of Hochman, we study the almost-Borel structure, i.e., the nonatomic invariant probability measures, of symbolic systems and surface diffeomorphisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a low entropy part is almost-Borel isomorphic to a Markov shift but that the remaining part is much more diverse, even for finite-to-one factors. However, we exhibit a new condition which we call "Bowen type" which gives complete control of those factors. This last result applies to and was motivated by the symbolic covers of Sarig. We find complete numeric invariants for Borel isomorphism of $C^{1+}$ surface diffeomorphisms modulo zero entropy measures; for those admitting a totally ergodic measure of positive (not necessarily maximal) entropy, we get a classification up to almost-Borel isomorphism.

A number of techniques have been developed to perturb the dynamics of $C^1$-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create local homoclinic orbits. These techniques have been crucial for the understanding of $C^1$ dynamics, but their most precise forms have mostly been shown in the dissipative setting. This work extends these results to volume-preserving and especially symplectic systems. These tools underlie our study of the entropy of $C^1$-diffeomorphisms in (arxiv:1606.01765). We also give an application to the approximation of transitive invariant sets without genericity assumptions.

These lecture notes focus on a recent result of Mike Hochman: an arbitrary standard Borel system can be embedded into a mixing Markov with equal entropy, respecting all invariant probability measures, with two exceptions: those carried by periodic orbits and those with maximal entropy. We discuss the corresponding notions of almost Borel embedding and isomorphism and universality. The main part of this paper is devoted to a self-contained and detailed proof of Hochman's theorem. We then explain how Katok's horseshoe theorem can be used to analyze diffeomorphisms with "enough" measures that are hyperbolic in the sense of Pesin theory, in both mixing and non-mixing situations. In the latter setting, new invariants generalizing the measures maximizing the entropy emerge.

We show that, for every positive real number h and every positive integer p, there exist oriented graphs G,G′ (with countably many vertices) that are strongly connected, of period p, of Gurevich entropy h, such that G is positive recurrent (thus the topological Markov chain on G admits a measure of maximal entropy) and G′ is transient (thus the topological Markov chain on G′ admits no measure of maximal entropy). We also show that any transitive topological Markov chain with infinite entropy carries uncountably many ergodic, invariant probability measures with infinite entropy. 1 Vere-Jones classification of graphs Definition 1 Let G be an oriented graph and let u, v be two vertices in G. We define the following quantities. • puv(n) is the number of paths u0 → u1 → · · · → un such that u0 = u and un = v; Ruv(G) is the radius of convergence of the series ∑ puv(n)z n. • fG uv(n) is the number of paths u0 → u1 → · · · → un such that u0 = u, un = v and ui 6= v for all 0 < i < n; Luv(G) is the radius of convergence of the series ∑ fG uv(n)z n. Definition 2 Let G be an oriented graph and V its set of vertices. The graph G is strongly connected if for all u, v ∈ V , there exists a path from u to v in G. The period of a strongly connected graph G is the greatest common divisor of (puu(n))u∈V,n≥0. The graph G is aperiodic if its period is 1. Proposition 3 (Vere-Jones [8]) Let G be an oriented graph. If G is strongly connected, Ruv(G) does not depend on u and v; it is denoted by R(G). If there is no confusion, R(G) and Luv(G) will be written R and Luv. In [8] Vere-Jones gives a classification of strongly connected graphs as transient, null recurrent or positive recurrent. The definitions are given in Table 1 (lines 1 and 2) as well as properties of the series ∑ puv(n)z n which give an alternative definition. Proposition 4 (Salama [7]) Let G be a strongly connected oriented graph. If G is transient or null recurrent, then R = Luu for all vertices u. Equivalently, if there exists a vertex u such that R < Luu, then G is positive recurrent.