O. Bandtlow
Title: Explicit resolvent bounds for transfer operators
Abstract: In this talk I will explain how to obtain explicit upper bounds for the resolvent R(z,L) of any compact operator L on a Hilbert space in terms of its singular values and the distance of z to the spectrum of L. I will then discuss applications of this estimate to obtaining explicit a priori error bounds for spectral approximations of transfer operators. Time permitting, I will also explain how to extend these results to the Banach space setting.

J. Bochi
Title: Exotic Lyapunov spectra

K. Burns
Title: The effect of Ricci flow on surfaces with no conjugate points
Abstract: This talk will describe joint work with Solly Coles and Dong Chen. We want to show that it is possible for conjugate points to appear when a surface with no conjugate points evolves under Ricci flow. This gives a negative answer to a question asked by Manning in a 2004 paper.

S. Cantrell
Title: The geometric joint spectrum and Manhattan manifolds
Abstract: Given a collection of matrices S in GL(d,C) the joint spectral radius is a number that quantifies the growth rate of products of matrices in S. Sert and Breuillard recently defined a multidimensional version of the joint spectral radius called the joint spectrum. In this talk we discuss a geometric analogue of the joint spectrum for group actions on metric spaces. To learn more about geometric joint spectra we introduce Manhattan manifolds: higher dimensional analogues of Manhattan curves. This is based on joint work with Eduardo Reyes and Cagri Sert.

S. Dyatlov,
Title: Ruelle zeta at zero for nearly hyperbolic 3-manifolds.
Abstract: For a compact negatively curved Riemannian manifold (Σ,g) theRuelle zeta function ζ (λ) of its geodesic flow is defined for Re(λ) > 1 as a convergent product over the periods T_λ of primitive closed geodesics ζ _R (λ)=∏_γ(1-e^{-λ T_γ}) and extends meromorphically to the entire complex plane. If Σ is hyperbolic (i.e. has sectional curvature -1), then the order of vanishing m _R(0) of ζ _R at λ=0 can be expressed in terms of the Betti numbers b_j(Σ). In particular, Fried proved in 1986 that when Σ is a hyperbolic 3-manifold, m_R(0)=4-2b₁(Σ). I will present a recent result joint with Mihajlo Cekic, Benjamin Kuster, and Gabriel Paternain: when dimΣ=3 and g is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely m_R(0)=4-b₁(Σ). This is in contrast with dimension 2 where m _R (0)=b₁(Σ)-2 for all negatively curved metrics. The proof uses the microlocal approach of expressing m_R(0) as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott--Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations.

A. Erchenko
Title: Local product structure of equilibrium states for geodesic flows.
Abstract: How do we define geodesic flows on CAT(0) spaces? In this talk we will concentrate on two settings: rank 1 nonpositively curved manifolds and flat surfaces with cone singularities of angles larger than 2π. We will discuss known results on the existence and uniqueness of equilibrium states for "nice" potentials for geodesic flows in those settings and their properties. Then, we will show how to obtain a local product structure in these settings using the non-uniform Gibbs property following an idea of Vaughn Climenhaga for the uniformly hyperbolic diffeomorphisms. This is based on the joint work with Benjamin Call, David Constantine, Noelle Sawyer, and Grace Work.

V. Kleptsyn
Title: Holder regularity of stationary measures.

H. Koivusalo
Title: Dynamical subsets in iterated function systems.
Abstract: The Poincare recurrence theorem tells us that in a measure preserving system, almost every point returns to a positive measure subset infinitely many times. The shrinking target problem is a natural follow-up: it asks about the size of the limsup set obtained from points with infinite returns to a shrinking sequence of subsets. There are many natural variants of the shrinking target problem, some of which are much harder than the original and require a vastly different toolkit. For example, one might be interested in the size of the liminf set of eventually always hitting points; or the size of a dynamical covering set, a limsup set of points hitting infinitely many target sets translated along orbits.

In this talk I will discuss some of these problems in the context of iterated function systems, finishing with a brand new result on dynamical covering sets (joint with Balazs Barany and Sascha Troscheit).

F. Ledrappier
Title: Entropy and measures at the boundary
Abstract: We describe equivariant families of measures on the boundary of the universal cover of a closed Riemannian manifold with negative curvature.
We discuss the associated entropy and its rigidity properties. The same formalism can describe
1.the Patterson-Sullivan family and the associated Burger-Roblin measure,
2. the Lebesgue family and the Liouville measure,
3. the harmonic measures and the drifted harmonic measures,
4. the Mohsen family giving the Rayleigh quotient and
5. the Gibbs-Patterson families.

J. Li
Title: On the dimension of limit sets on the real projective plane via stationary measures
Abstract: I will present a dimension jump result of limit sets on RP^2 for representations of surface groups in SL(3,R). For Anosov representations, we prove the equality between the Hausdorff dimension and the affinity exponent. In particular, it exhibits a dimension jump under perturbation.
The key tool is to study the stationary measures of finitely supported random walks on SL(3,R). We show the Hausdorff dimensions equal the Lyapunov dimensions under certain assumption. This is based on the joint work with Wenyu Pan, Disheng Xu and partially with Yuxiang Jiao.

F. Naud
Title: Resonances of random Schottky groups and strong convergence
Abstract: The purpose of this talk is to explain how some tools coming from random matrices and free probability can be used to study spectral gaps of random covers of Schottky surfaces. Joint work with Irving Calderon and Michael Magee.

Y. Pesin
Title: The Essential Coexistence Phenomenon in Hamiltonian Dynamics
Abstract: A volume preserving dynamical systems with discrete or continuous time on a compact smooth manifold M is said to exhibit essential coexistence if M can be split into two invariant disjoint Borel subsets A and B of positive volume -- the chaotic and regular regions -- such that:
1) for a.e. x in the set A the Lyapunov exponents at x are all nonzero (except for the Lyapunov exponents in the flow direction in the case of continuous time);
2) for every x in the set B the Lyapunov exponents at x are all zero;
3) the restriction of the system on the set A is ergodic (Bernoulli).
There are two types of essential coexistence: type I, when the set A is open (mod0) and dense and type II, when the set B is open.
I will discuss some general results, conjectures, and present some examples of systems with both discrete and continuous time which exhibit essential coexistence of type I. Finally, I will outline a construction of a Hamiltonian flow with essential coexistence of type I (based on a recent joint work with J. Chen, H. Hu, and Ke Zhang).

J. Slipantschuk,
Title:Distribution of resonances for Anosov maps
Abstract: I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.

D. Thompson
Title: Specification and strong positive recurrence for flows on complete metric spaces
Abstract: We extend Bowen’s approach to thermodynamic formalism to flows on complete separable metric spaces. We define a suitable notion of specification in this setting, which gives uniform transition times for orbit segments which start and end in a fixed closed ball (with the transition time allowed to be larger if the ball is larger). The key point, particularly for the existence of an equilibrium state, is a Strong Positive Recurrence (SPR) assumption defined at this level of generality. As one application, we establish that for a sufficiently regular potential with SPR for the geodesic flow on a geometrically finite locally CAT(-1) space, there exists an ergodic Gibbs measure. This measure is finite, and is the unique equilibrium state. This is joint work with Vaughn Climenhaga and Tianyu Wang.

G. Tiozzo
Title: The Poisson boundary of hyperbolic groups without moment conditions

M. Tsujii
Title: Virtually expanding dynamics
Abstract: We introduce a class of discrete dynamical systems that we call virtually expanding. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron–Frobenius operator is quasi-compact on a Sobolev space of positive order for such a class of dynamical systems.

M. Urbanski
Title: Escape Rates, Surviving Equilibrium States and Hausdorff Dimension for Open Dynamical Systems.

P. Vytnova
Title: Dimension function of the Lagrange and Markov spectra.
Abstract: I will discuss an approach for computing the Hausdorff dimension of an intersection of
the classical Lagrange and Markov spectra with half-infinite ray d(t) = dim(M ∩ (−∞,t)), that
allows to plot a graph of the function d(t) with high accuracy. The talk is based on a joint work
with Carlos Gustavo Moreira and Carlos Matheus Santos and Mark Pollicott.

D. Xu
Title: On the dimension theory of random walks and group actions by circle diffeomorphisms.
Abstract: We establish new results on the dimensional properties of measures and invariant sets associated to random walks and group actions by circle diffeomorphisms. This leads to several dynamical applications. Among applications, we strengthen a recent breakthrough of Deroin-Kleptsyn-Navas by proving that the exceptional minimal (Cantor) set of a finitely generated group of real-analytic circle diffeomorphisms must have Hausdorff dimension less than one. Moreover, if the minimal set contains a fixed point of multiplicity k+1 of a diffeomorphism of the group, then its Hausdorff dimension must be greater than k/(k+1). These results generalize classical results about Fuchsian group actions on the circle to non-linear settings. This is a joint work with Weikun He and Yuxiang Jiao.

A.Zelerowicz
Title: Lorentz gases on quasicrystals
Abstract: The Lorentz gas was originally introduced as a model for the movement of electrons in metals.
It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers S (atoms of the metal) with elastic collisions at the boundaries ∂ S. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Trevino. We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane.

Z. Zhang,
Title: Geometric properties of partially hyperbolic measures and applications to measure rigidity
Abstract: We give a geometric characterization of the quantitative joint non-integrability, introduced by Asaf Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of uu-states and the existence of physical measures. This is a joint work with Alex Eskin and Rafael Potrie.