Talk abstracts

Below are the talk abstracts for the SHARP conference - the full schedule is available on a separate page.

Longer talks

Alves: Linear response for skew-product maps

We study linear response for families of skew-product dynamical systems with mean contracting fibres. Using a sectional transfer operator acting on families of probability measures along the fibres, we describe invariant measures of the skew-product through sample measures over the base dynamics, independently of the invertibility of the base map. Under general assumptions we prove existence, uniqueness and differentiability of invariant sample measures with respect to system parameters. As applications, we obtain linear response for some variations of baker maps and for physical measures of solenoidal attractors with intermittency, a class of partially hyperbolic systems beyond the reach of standard transfer operator methods.

Baker: Thermodynamic formalism and Fourier decay

An important function giving arithmetic and geometric information about a measure is its Fourier transform. Determining decay rates for the Fourier transform has consequences in many areas of mathematics, such as proving the existence of normal numbers in the support of the measure. In this talk I will explain how spectral properties of transfer operators can be used to prove that stationary measures associated to iterated function systems have polynomial Fourier decay. This talk will be based upon some joint works with Khalil and Sahlsten.

Burns: Uniqueness of the measure of maximal entropy for geodesic flows on surfaces with caps

The class of surfaces in this talk was introduced in the 1980s by Donnay in order to exhibit a smooth Riemannian metric on the two sphere with ergodic geodesic flow with respect to the smooth Liouville measure. Recent joint work with Todd Fisher and Rachel McEnroe has shown that the geodesic flows for these surfaces have unique (and therefore ergodic) measures of maximal entropy.

Butt: Monotonicity of topological entropy along the Ricci flow

The normalized Ricci flow (NRF) is a natural deformation on the space of Riemannian metrics which “simplifies” the curvature. For instance, let (M, g) be a closed surface of variable negative curvature; then the normalized Ricci flow g_t starting from g converges to a metric of constant negative curvature as t tends to infinity. In this setting, Manning showed that the topological entropy of the geodesic flow of g_t is strictly decreasing in t. He also asked if an analogous result holds in higher dimensions in a neighborhood of a hyperbolic metric. The main result of this talk, joint with Tristan Humbert and Alena Erchenko, is that the answer is yes. In addition to geometric and dynamical techniques, the proof uses tools from microlocal analysis.

Fraser: Subtle horoball asymptotics recording parabolics

Consider a non-elementary geometrically finite Kleinian group with at least one parabolic element. The parabolic fixed points form a countable dense subset of the limit set and to this set of parabolic points we may associate a standard set of horoballs. The size of each horoball can be thought of as analogous to the 'cost' 1/q associated with a rational number p/q used in, for example, Diophantine approximation. I will discuss the problem of counting horoballs of a given size, with particular emphasis on local variants. This is based on joint work with Liam Stuart.

Gekhtman: Linearly growing injectivity radius in negatively curved manifolds with small critical exponent

Let X be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice \Gamma. For instance, X could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let H be a discrete subgroup of isometries of X with critical exponent (exponential growth rate) strictly less than half of the growth rate of \Gamma. We show that the injectivity radius of X/H grows linearly along almost every geodesic in X (with respect to the Patterson-Sullivan measure on the Gromov boundary of X). The proof will involve an elementary analysis of a novel concept called the "sublinearly horosherical limit set" of H which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups.

Ghazouani: Martin-Baillon Spectrum of SL(2,R)-representations

Representations of surface groups in an arbitrary Lie group G appear naturally in the study of geometric structures on surfaces. The case G = SL(2,R) is of particular importance due to its relevance to hyperbolic geometry. In this talk we introduce a new dynamical invariant of such representations, which we call its spectrum; it is a fractal set which we believe to capture most geometric features of the representation. We will mostly concentrate on the particular case of the punctured torus, in order to keep the talk as elementary as possible.

Gogolev: Deformation rigidity in proximity of de la Llave examples

De la Llave’s examples are certain skew-product Anosov diffeomorphisms that display a number of very interesting properties. In particular, they form an isospectral family that does not belong to a fixed smooth conjugacy class — a property reminiscent of the “you cannot hear the shape of a drum” counterexamples to Mark Kac’s question. I will explain that nevertheless a generic perturbation of a De la Llave example does not admit such isospectral deformations. Joint work with Martin Leguil.

Jurga: Exceptional sets in parameter spaces of dynamical systems

Many dynamical systems arise naturally as parametrised families. One can then study the associated parameter space and ask how qualitative properties of the dynamics are reflected geometrically within it. In this talk I will discuss three examples where classification problems lead to intricate subsets of parameter space: the Mandelbrot set in complex dynamics, the Rauzy gasket arising from interval exchange transformations, and a tiling structure associated with the topological classification of a family of self-similar sets. Although these examples arise in very different settings, they exhibit striking parallels in how parameter spaces organise qualitative changes in the underlying dynamics.

Kao: Manhattan curve and the correlation number of cusped Hitchin representations

In this talk, we discuss the Manhattan curve and its relationship with the correlation number of cusped Hitchin representations. Motivated by earlier work of Richard Sharp, we show that several of the phenomena he observed extend to the setting of geometric structures with cusps. This is joint work with Giuseppe Martone.

Kosloff: Stationary factors of chaotic systems away from equilibrium

There are several notions of chaos for diffeomorphisms. A notable one for conservative diffeomorphisms, called statistical chaos, is that one can simulate a random stochastic process, say a fair coin tossing process, from the given dynamical system with respect to the volume measure.

In a previous work with Terry Soo we noticed that all dissipative diffeomorphisms, even very structured ones like Morse-Smale systems, are chaotic in this sense and as a result this notion is not meaningful in this case. We will discuss some recent findings on a more refined notion called finitely chaos and demonstrate using two large classes of chaotic systems away from equilibrium how this notion behaves much better. Based on joint work with Terry Soo.

Kucherenko: Freezing Phase Transitions for Higher-Dimensional Shifts

We discuss the existence of freezing phase transitions in the setting of higher-dimensional shift spaces. We focus on the type of potentials that trigger a freezing phase transition where the support of the resulting ground state is an arbitrary predetermined subsystem. To contrast this result, we give sufficient conditions on the potential that guarantee that the system never freezes. This is joint work with Jean-René Chazottes and Anthony Quas.

Ledrappier: Exact-dimension of Furstenberg measures

We consider a probability measure with finite support on a real algebraic semi-simple group with finite center. The Furstenberg measure is exact dimensional (follows from Rapaport 21). We explain the ingredients of the entropy/exponent formula for the dimension in the general case. This is a joint work with Pablo Lessa.

O’Hare: Effective Equidistribution for Contact Anosov flows in Dimension Three

A classical theorem of R. Bowen states that the family of normalized measures over geodesic packets in hyperbolic surfaces equidistributes towards the measure of maximal entropy. In a joint work with Asaf Katz, we quantify this equidistribution in the general setting of contact flows in dimension 3, recovering an exponential equidistribution rate. When these periodic orbit measures are weighted by a potential function, we show that they exponentially equidistribute towards the appropriate equilibrium state. Our estimate is based on Zeta function analysis approach of Pollicott-Sharp together with Dolgopyat’s method. We also recover weaker (polynomial) bounds for general transitive Anosov flows.

Pollicott: Richard Sharp: some results suggested by his work and life

I will discuss some recent results which were suggested by work of Richard Sharp. These include (time permitting) questions related to: the Schwartzman cocycle and entropy; modular knots; and Livsic-type theorems (the latter being joint work with Richard Sharp).

Sert:Growth indicator and translation cone for Gromov hyperbolic groups

We first introduce a class of metric-like functions on hyperbolic groups, called hyperbolic metric potentials. This is a class of functions general enough to include word-metrics, quasi-morphisms, and the fundamental weights of Anosov representations. Then, given a tuple (f1,...,fd) of such functions, we introduce the notion of translation cone, an analogue of the limit cone introduced in the setting of linear algebraic groups by Benoist in the 90s. We establish analogues of Benoist's results as well as additional hyperbolic facts on this cone. We then turn to a more precise asymptotic analysis: counting. We introduce the analogue of the growth indicator function, introduced in early 2000s by Quint again in the linear setting. We show that this function is always strictly concave and C1, this generalizes several results of Quint, Sambarino, Kim-Oh-Wang, etc. Finally, we relate this function to a multi-dimensional generalization of the Manhattan curve, which is a curve of Poincaré exponents, introduced in dimension 2 by Burger in the 90s, and recently studied in our more general setup by Tanaka, and Cantrell--Tanaka. Joint work with Stephen Cantrell and Eduardo Reyes.

Stadlbauer: Non-expanding maps, Martin boundaries and positive drift

In this talk, we study random walks on hyperbolic groups with stationary increments using the thermodynamic formalism of skew-product systems \[T:\Sigma \times X \to \Sigma \times X, \qquad (\xi,x)\mapsto(\theta \xi,\kappa_\xi(x)), \qquad \varphi:\Sigma\to\mathbb{R},\] where $\Sigma$ and $X$ are compact metric spaces, $T$ is uniformly expanding, $\kappa$ maps $\Sigma$ continuously into the homeomorphisms of $X$, and $\varphi$ is H\"older continuous.

Under a suitable local backward contraction condition on average, a Perron--Frobenius--Ruelle theorem holds, which allows us to obtain synchronization and almost sure invariance principles for the skew product.

Verifying this contraction for $X=\partial G$ relies on properties of the Martin boundary. In particular, it allows us to prove a Karlsson--Margulis--type ergodic theorem and the positivity of the drift, extending a result of Kaimanovich to the setting of stationary increments. As a consequence, we obtain an almost sure invariance principle for the drift in $G$.

Joint work with Gil Astudillo and Sara Bispo.

Short talks

Anttila: A variational principle for the Hausdorff dimension of non-linear carpets

I will discuss dimension theory of non-linear carpets, which are a family of attractors of iterated function systems that are both non-linear and non-conformal. I will describe how the Hausdorff dimensions of these carpets can be approximated arbitrarily well by Hausdorff dimensions of ergodic measures supported on the attractor. As an application, one can show that the set of badly approximable points on a non-linear carpet has full Hausdorff dimension. The talk is based on joint work with Jonathan Fraser and Henna Koivusalo.

Baumgartner: Orbit Counting in Conjugacy Classes

We consider an orbit counting problem for discrete group actions on negatively curved spaces where we restrict to counting within a single conjugacy class. This problem and related ones were previously studied by a number of authors including Huber, Guillopé, Douma, Kenison-Sharp, Broise-Alamichel-Parkkonen-Paulin, Parkkonen-Paulin and Honaryar. In joint work with Pollicott, we introduce a generalisation of the Cannon coding for hyperbolic groups which allows us to establish asymptotics for convex co-compact actions on CAT(-1) spaces.

Khodaeian Karim: Adiabatic Invariant actions for Partially Integrable Systems

This is a joint work with Konstantinos Kourliouros and Dmitry Turaev. We introduce action variables for partially integrable systems. We prove that for slow-fast partially integrable systems, these actions are adiabatic invariants of motion, and give rise to integrals for the averaged system, provided that the system is ergodic, for each fixed value of the slowly varying parameters, on the common levels of their integrals. We would also discuss the averaged system, when there is no ergodicity assumption. This work raises questions that stand in the crossroads of Ergodic Theory, Symplectic Topology, and Thermodynamics.

Tumarkin: Self-similar straight-line flows on abelian covers of translation surfaces

I will discuss an application of hyperbolic dynamical tools to the study of a zero entropy system, namely the straight-line flow on an infinite abelian cover of a translation surface. We focus on the case when this flow is self-similar, i.e. it is renormalised by a pseudo-Anosov map.

By studying a family of twisted transfer operators for the pseudo-Anosov map, we deduce several dynamical results about the flow and its invariant measures, such as an asymptotic expansion for ergodic integrals. As a byproduct we recover some known results, including a formula for the Hausdorff dimension of the invariant measures recently found by Berk, Frączek, Kotlewski and Trujillo. This is joint work with Mauro Artigiani, Roberto Castorrini and Davide Ravotti.

Xu: Dimensions of orbital sets in complex dynamics

We investigate the box dimension of orbital sets in complex dynamics. In particular, we are interested in the relationship between the box dimension of the orbital set and the box dimension of the associated Julia set for any rational map with degree at least 2 defined on the Riemann sphere. Joint work with Jonathan Fraser.