Schedule for: 23w5095 - Thermodynamic Formalism for Geodesic Flows

We will give an overview of different approaches to obtaining asymptotic estimates for counting closed geodesics on negatively curved manifolds (e.g. using the Margulis approach or using zeta functions). The will also discuss generalizations (e.g., no conjugate points) and equidistribution results

Bowen proved uniqueness of equilibrium states for expansive systems with specification and suitably nice potentials. Later, Climenhaga and Thompson generalized this argument to abstract systems where all of the properties Bowen requires hold only non-uniformly and at a fixed scale. When specification holds at all scales, their result was later shown to imply various mixing properties. I will give an overview of the machinery and some applications in this simpler setting, which includes the geodesic flow on flat surfaces with conical singularities. If time permits, I will also discuss a recent result by Pacifico, Yang, and Yang, which generalizes the settings in which their machinery applies.

In this talk, we will discuss a thermodynamic formalism approach to studying the geometry of deformation spaces of geometric structures on surfaces such as Teichm\"uller spaces (hyperbolic structures) and higher Teichm\"uller spaces (convex projective structures, etc.) In particular, we will discuss/survey results, including orbital counting, correlation number, pressure metric, and Thurston's asymmetric metric. When the geometric structures contain cusps, these results are new and are joint works with Harry Bray, Dick Canary, and Giuseppe Martone.

Two or three micro talks

Four or five 10-minute micro-talks; or more problem session, as needed.

We will have a panel aimed at early career participants. All are welcome.

This lecture will be primarily based on my book "Open Dynamical Systems: Statistics, Geometry, and Thermodynamic Formalism" joint with Tushar Das, Giulio Tiozzo, and Anna Zdunik. The starting dynamical system will be given by any countable alphabet finitely primitive subshift of finite type and the invariant Gibbs state of a 1-cylinder summable Holder continuous potential $\varphi$. The holes defining open systems will be then formed by collections of open sets with appropriately constrained thin boundaries. They will naturally lead to singularly perturbed transfer operators associated to the potential $\varphi$. The talk will develop conditionally invariant measures and surviving sets. The surviving variational principle, the existence and uniqueness of surviving equilibrium states, the asymptotic of perturbed leading eignevalues, and escape rates when holes approach a given hole will be discussed. The logarithms of leading perturbed eigenvalues will turn out to be equal to topological pressures defined be means of surviving variational principle. Also stochastic properties of surviving equilibria such as exponential decay of correlation, the Central Limit Theorem, the Law of Iterated Logarithm, and the Almost sure Invariance Principle will be discussed. All these objects will be produced and studied in our dynamical setting by means and applications of the Keller-Liverani Perturbation Theorem. Applications to “real” systems will be also discussed. It will be shown that in such a context in Euclidean spaces, virtually all Euclidean balls taken as holes give rise to "controlled" open systems. In particular, these holes need not be dynamically defined, for example as unions of cylinders of the same length. For conformal systems, the asymptotic of Hausdorff dimension of surviving sets will be also computed.

On the geodesic flow of a manifold of pinched negative curvature, there is a natural quasi-product construction of equilibrium states in terms of Patterson-Sullivan measures, which are measures that are defined geometrically on the visual boundary of the universal cover. For a locally CAT(-1) space with a non-constant potential, the construction was not as clear due to discrepancies arising from non-uniqueness of extensions of geodesic segments. I will describe a joint work with Daniel Thompson showing that these discrepancies introduce only uniformly bounded errors in the construction, so that Coornaert’s ideas from the Gromov hyperbolic setting can still be applied to obtain an invariant measure, and I will discuss what this means for the thermodynamic formalism.

Several stochastic processes are defined on the hyperbolic plane H^2. For instance, one can consider a Brownian motion, or a discretized version thereof, when one performs a random walk on the group of isometries of H^2. It is a recurring question, going back to Furstenberg, Guivarc’h, Ledrappier, Kaimanovich, and others, whether the measures obtained from the random walks coincide with measures of geometric origin, such as the Lebesgue measure. One is also interested in the Hausdorff dimension of the harmonic measure. In the talk, we will explore some of the classical results in this area, as well as discussing two methods to prove the singularity of harmonic measure: one is related to cusp excursions for the geodesic flow, the other one to the so-called “fundamental inequality” between entropy and drift. We will present some recent progress which uses these methods. Based on joint works with J. Maher, V. Gadre, I. Gekhtman, A. Randecker, and P. Kosenko.

This talk will review recent progress in proving existence of equilibrium states for several classes of potentials for dispersing billiards without corner points. These equilibrium states include the measures of maximal entropy for both the billiard maps and the associated billiard flows. The main technique involves the construction of Banach spaces adapted to the potentials on which the associated transfer operators can be studied.

In this talk, we prove the uniqueness of the measure of maximal entropy for the geodesic flow on surfaces of genus at least two when restricted to the set of minimal geodesics. This generalizes the recent result by Climenhaga-Knieper-War for surfaces without conjugate points. We also prove that this measure can be constructed via Patterson-Sullivan measures. This is based on a joint work with Gerhard Knieper.