# Titles and Abstracts

Title: The fractional susceptibility function for the quadratic family

Abstract: For $t$ in a set $\Omega$ of positive measure, maps in the quadratic family $f_t(x)=t-x^2$ admit an SRB measure $\mu_t$. On the one hand, the dependence of $\mu_t$ on $t$ has been shown (by Benedicks, Schnellmann, and the speaker) to be no better than $1/2$ Hölder, on a subset of $\Omega$, for $t_0$ a suitable Misiurewicz--Thurston parameter. On the other hand, the susceptibility function $\Psi_t(z)$, whose value at $z=1$ is a candidate for the derivative of $\mu_t$ with respect to $t$, has been shown by Jiang and Ruelle to admit a holomorphic extension at $z=1$ for $t=t_0$. Our goal is to resolve this paradox. For this, we introduce and study a fractional susceptibility function. (Joint work in progress with Daniel Smania.)

Snir Ben Ovadia (Weizmann Institute) SHORT TALK

Title: Generalized SRB measures, physical properties, and thermodynamic formalism of smooth hyperbolic systems

Abstract: We generalize the notion of SRB measures and characterize their existence and uniqueness; and study their ergodic, thermodynamic, and physical properties. We show that there exists a generalized SRB measure (GSRB for short) if and only if there exists some unstable leaf with a positive leaf volume for the set of hyperbolic points. This condition is called the leaf condition. We show that the leaf condition also implies that the Gurevich pressure of the geometric potential is 0, and characterize all GSRBs as equilibrium states of the geometric potential, which gives the analogue to the entropy formula. Every finite GSRB is an SRB, and the finiteness of GSRBs is characterized by modes of recurrence of the geometric potential. We define a set of positively recurrent points, and characterize positive recurrence of the geometric potential by the leaf condition for the said points. We show the uniqueness of GSRBs on each ergodic homoclinic class, and that each ergodic component of a GSRB is a GSRB. We show physical properties for GSRBs (w.r.t. the ratio ergodic theorem, and by distributions). In our setup, $M$ is a Riemannian, boundaryless, and compact manifold, with $\dim M\geq 2$; $f\in \mathrm{Diff}^{1+\beta}(M), \beta>0$.

Keith Burns (Northwestern)

Title: Openness of accessibility for partially hyperbolic diffeomorphisms with three dimensional centre

Abstract: Consider partially hyperbolic diffeomorphisms with an invariant splitting into three bundles – unstable, center, and stable. Didier showed that if the center bundle is one dimensional, then the property of accessibility is open, i.e. it persists under small perturbations of the diffeomor- phism. More recently Avila and Viana showed that same is true when the center bundle is two dimensional. The talk will describe an effort to adapt Avila and Viana’s technique to show that accessibility is still open when the center bundle is three dimensional. This is joint work with Jana Rodriguez Hertz and Raul Ures.​

Italo Cipriano (PUC, Santiago) SHORT TALK

Title: Approximating integrals with respect to stationary probability measures

Abstract: From a dynamical approach, the problem of approximation of integrals with respect to stationary probability measures is analogue to the problem of approximation of integrals with respect to the Lebesgue measure studied by Jenkinson and Pollicott in [”A dynamical approach to accelerating numerical integration with equidistributed points.” Proceedings of the Steklov Institute of Mathematics 256.1 (2007): 275-289]. In this talk, I will enunciate a theorem and give an algorithm that yields accelerated approximations of integrals with respect to stationary probability measures. Finally, I will explain how this result can be applied to estimate the Wasserstein distance between certain stationary probability measures. This is joint work with Natalia Jurga.

Vaughn Climenhaga (Houston)

Title: Bowen-Margulis measure for certain billiards and geodesic flows

Abstract: Two distinct constructions of the unique measure of maximal entropy (MME) for a uniformly hyperbolic system were given by Bowen and by Margulis. This talk will review progress in extending versions of both approaches to non-uniformly hyperbolic systems such as the Bunimovich stadium billiard and geodesic flows without conjugate points, and will highlight some of the remaining open problems.

Rhiannon Dougall (Bristol)

Title: Countable group extensions of subshifts of finite type

Abstract: A particular family of countable Markov shifts is given by "group extensions" of subshifts of finite type, or "skew products with countable groups". Such a system $X$ is given by a countable group $G$, an alphabet $\{0,\ldots, k\}\times G$, and a transition matrix that is invariant under the natural $G$ action, and so factors on to a subshift of finite type $\Sigma$. We consider potentials $r:X\to \mathbb{R}$ that are invariant under $G$. There have been a number of developments in recent years on questions relating the Gurevic pressure of $r$ on $X$ to the pressure of the factor $r$ on $\Sigma$; and on questions of spectral radius of transfer operators. These have applications to regular covers of Anosov flows and growth of periodic orbits. We will highlight some interesting features of this universe.

Godofredo Iommi (PUC, Santiago)

Title: Escape of mass and entropy at infinity

Abstract: In the context of countable Markov shifts, I will present a result that relates the escape of mass, the measure theoretic entropy and the entropy at infinity of the system. This relation has several consequences. For example, that the entropy map is upper semi-continuous. This is joint work with Mike Todd and Anibal Velozo.

Michael Jakobson (Maryland)

Title: Hénon-like maps revisited

Abstract: We prove positivity of the measure of parameters with strong mixing properties for some Hénon-like models with jacobian less than one. We combine several methods: parameter exclusion, adapted coordinates, Palis-Yoccoz implicit formalism.

Anders Johansson (Gävle) CANCELLED

Title: Conditions for uniqueness of g-measures

Abstract: We will survey some joint results with Anders Öberg and Mark Pollicott.

Natalia Jurga (Surrey) SHORT TALK

Title: Box dimensions of exceptional self-affine sets in $R^3$

Abstract: In the 1980s, Falconer established a formula for the Hausdorff and box dimension of a "typical" self-affine set. However, following recent results of Bárány, Hochman and Rapaport, it has emerged that in the planar setting this formula actually holds for all self-affine sets, outside of a small family of exceptions. This family of exceptions includes self-affine sets which have been generated by generalised permutation matrices. In this talk we will discuss the box dimensions of self-affine sets which have been generated by 3x3 generalised permutation matrices. This is joint work with Jonathan Fraser.

Lien-Yung Kao (Chicago)

Title: Pressure Metrics and Manhattan Curves for Teichmüller Spaces of Punctured Surfaces

Abstract: Thurston pointed out that one can use variations of lengths of closed geodesics on hyperbolic surfaces to construct a Riemannian metric on the Teichmüller space. For closed surfaces cases, Wolpert proved this Riemannian metric is indeed the Weil-Petersson metric. McMullen proposed a thermodynamic formalism approach to this Riemannian metric, and called it the pressure metric. In this talk, I will discuss how to extend this dynamics construction to non-compact finite area hyperbolic surfaces. If time permitted, I will also discuss relations between the pressure metric and Manhattan curves.

Tom Kempton (Manchester)

Title: Dimension for Bernoulli Convolutions

Abstract: Bernoulli convolutions are a simple family of fractal measures with overlaps. The study of the dimension theory of Bernoulli convolutions goes back to the 1930s, yet in many cases of algebraic parameters the dimension of the convolution remains unknown. In this talk we describe some higher dimensional structures and an associated dynamical system that allows one to compute the dimension.

Tamara Kucherenko (City College of New York)

Title: Structure of measures of maximal entropy for topological suspension flows

Abstract: We establish the existence of a suspension flow with continuous roof function which satisfies the following property. The set of measures of maximal entropy for the flow consists precisely of measures which maximize entropy on a prescribed invariant subset on the base. This result has a number of corollaries on how the set of measures of maximal entropy for the flow can be bad, even over a very nice space such as the full shift. (Joint work with D. Thompson.)​

François Ledrappier (CNRS)

Title: Rigidity in negative curvature

Abstract: I will survey some results and problems related to recognizing compact manifolds with negative curvature that are locally symmetric spaces.

Carlangelo Liverani (Rome 2 "Tor Vergata")

Title: Fast slow partially hyperbolic systems

Abstract: Both fast-slow systems and partially hyperbolic systems are at the center of intense studies. Yet few attempts exist to merge the techniques used in this two fields in order to obtain a precise description of the long time behaviour of such systems. I will describe some preliminary results in this direction.

Matthew Nicol (Houston)

Title: Some results on large deviations and central limit theorems for sequential and random systems of intermittent maps

Abstract: We give polynomial large deviations estimates for sequential and random systems of intermittent-type maps and discuss the role of centering in quenched central limit theorems for random dynamical systems of intermittent- type maps. (Joint with Felipe Perez Pereira.)

Frédéric Paulin (Paris-Sud)

Title: Logarithm laws for equilibrium states in negative curvature

Abstract: Let M be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure $m_F$ associated with a potential $F$. We study the $m_F$-almost sure asymptotic penetration behaviour of locally geodesic lines of M into small neighbourhoods of closed geodesics. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objects. As an arithmetic consequence, we give almost sure Diophantine approximation results of real numbers by quadratic irrationals with respect to general Hölder quasi-invariant measures. (Joint work with Mark Pollicott.)

Yakov Pesin (Penn State)

Title: Reflections on Viana's conjecture: SRB measures for systems with non-zero Lyapunov exponents

Abstract: In his plenary lecture at the ICM in Berlin in 1998, Viana proposed a conjecture that a diffeomorphism with non-zero Lyapunov exponents at Lebesgue almost every point should admit an SRB measure. In this talk I will describe some recent breakthrough results on existence of SRB measures for some general hyperbolic systems.

Mark Piraino (Victoria) SHORT TALK

Title: Matrix equilibrium states

Abstract: In classical thermodynamic formalism equilibrium states maximize entropy plus the integral of a function over the space of invariant measures. In analogy matrix equilibrium states maximize entropy plus the top Lyapunov exponent of a matrix co-cycle. In this talk I will discuss matrix equilibrium states for 1-step co-cycles over a full shift. I will present a new way of constructing these measures which is based on the use of a transfer operator. In addition we will see how ergodic properties of these measures can be deduced from spectral properties of the transfer operator.

Anke Pohl (Bremen)

Title: Thermodynamic formalism and cohomology for resonance states of Laplace-Beltrami operators

Abstract: Since several years it is known that certain discretizations for the geodesic flow on hyperbolic surfaces of finite area allow to provide a dynamical characterizations of Maass cusp forms and a transfer-operator-based construction of their period functions. An important ingredient for these results is the characterization of Laplace eigenfunctions in parabolic cohomology by Bruggeman-Lewis-Zagier.
We discuss an extension of these results to Hecke triangle surfaces of infinite area and Laplace eigenfunctions that are more general than Maass cusp forms. This is joint work with R. Bruggeman.

Hans Henrik Rugh (Paris-Sud)

Title: CLT in odometer translations and matrix co-cycles

Abstract: Consider the sum-of-digit function $s(n)$ on positive integers (e.g. in base 10 one has: $s(723)=7+2+3=13$) and for $a>0$ the distribution of the sum-of-digit difference $s(n+a)-s(n)$ when $n$ is picked uniformly in $[0,N]$. We show that when $N$ and a goes to infinity in an "essential" way the problem verifies a uniform CLT (so we also have e.g. a Berry-Esseen estimate for the convergence rate). The proof works by mapping the characteristic function of a distribution to a study of an analytic matrix co-cycle and obtain uniform analytic bounds for the characteristic exponents of this co-cycle. (Joint work with Jordan Emme.)

Federico Rodriguez Hertz (Penn State)

Title: Some consequences of exponential mixing

Abstract: In this talk I shall review some basic consequences of exponential mixing and and other rates of mixing and try to state some open problems. This is an ongoing project.

Omri Sarig (Weizmann Institute)

Title: Local limit theorems for inhomogeneous Markov chains

Abstract: An inhomogeneous Markov chain is a Markov chain $X_1,X_2,\ldots$ whose set of states and transition kernels are allowed to vary in time. In his PhD, R. Dobrushin proved a central limit theorem for sums of the form $S_N=f_1(X_1,X_2)+ \cdots +f_N(X_N,X_{N+1})$. I will discuss the local limit theorem in case $f_i$ are uniformly bounded and $\{X_i\}$ is uniformly elliptic. (Joint with D. Dolgopyat.)

Ofer Shwartz (Weizmann Institute) SHORT TALK

Title: Eigenfunctions and eigenmeasures of the Ruelle operator

Abstract: A potential function over a Markov shift with infinite number of states can be classified as either recurrent or transient. It is known that in the recurrent case, the eigenmeasure and the eigenfunction of the Ruelle operator exist and are unique and the eigenmeasure is conservative. However, if the potential is transient, then there exist eigenmeasures and eigenfunctions which are not necessarily unique and the eigenmeasures are totally dissipative. In this talk, we will show that these eigenmeasures and eigenfunctions can be fully characterized by a suitable Martin boundary and present a duality between the two.

Karoly Simon (Budapest University of Technology and Economics)

Title: The computation of the dimension of some non-conformal attractors with the transversality method

Abstract: Consider a one-parameter family of self-conformal (with overlapping cylinders) or self-affine sets. In many cases, we cannot compute the dimension of the individual attractors. In these cases, we may hope that we can compute the dimension at least for a typical member of this family. The method most commonly used for this purpose is called transversality method. In my talk, I will give a short account of this method. In particular, for the self-affine attractors and measures, T. Jordan, M. Pollicott and I introduced a version of transversality method. I would like to present a variant of this method which was used very recently by D.J. Feng and I for the computation of the dimension of some non-conformal attractors.

Mariusz Urbanski (North Texas)

Title: Dimension Spectrum of Conformal Iterated Function Systems

Abstract: We will define conformal iterated function systems $S$ over a countable alphabet $E$ and theirlimit sets (attractors) $J_E$. We will discuss the formula for the Hausdorff dimension of this limit set, commonly referred to as a version of Bowen's formula, involving topological pressure. The main focus will be on the set $$Sp(E)=\{HD(J_F): F\subset E\},$$ called the dimension spectrum of the system $S$. I will prove that always $$Sp(E)\supset (0,\theta_E),$$ where $\theta_E$ is the finiteness parameter of $S$ (will be defined). I will also construct a system for which $Sp(E)$ is a proper subset of $(0,HD(J_E)]$. I will then discuss the property that $$Sp(S)=(0,HD(J_E)],$$ called the full spectrum dimension property. In particular, I will consider the conformal iterated function systems and their various subsystems, generated by real and complex continued fraction algorithms, and will show that many of them (subsystems) enjoy the full spectrum dimension property.

Anibal Velozo (Yale) SHORT TALK

Title: Pressure and pressure at infinity of countable Markov shifts

Abstract: In this talk I will explain some recent development in the ergodic theory of countable Markov shifts (CMS for short). For finite entropy CMS I will introduce the pressure at infinity of a potential and explain its relationship with the escape of mass and pressure of measures. If time permits I will explain how the methods involved allow to prove the upper semi-continuity of the pressure map (under suitable assumptions on the potential, in particular, potentials with finite pressure), and other applications.

Howie Weiss (Georgia Tech)

Title: Respiratory disease transmission in airplane cabins

Abstract: With over three billion airline passengers annually, the inflight transmission of infectious diseases is an important global health concern. Over a dozen cases of inflight transmission of serious disease have been documented, but, despite the sensational media stories and anecdotes, the true risks of transmission are largely unknown. Studies of Severe Acute Respiratory Syndrome (SARS) and pandemic influenza (H1N1p) transmission on airplanes indicate that air travel can serve as a conduit for the rapid spread of newly emerging infections and pandemics. It is believed that the movements of passengers and crew facilitate disease transmission.

We report on the results of our FlyHealthy research study, where we attempt to quantify the rates and routes of transmission of infectious diseases during air travel. Goals include characterizing the microbial communities in the cabin (in particular those in the air and on common touch surfaces), quantifying transmission opportunities, and constructing a data-driven, dynamic network transmission model of droplet mediated respiratory diseases.