Titles and abstracts


Name
Vaughn Climenhaga
Institution
University of Houston
Title:

 Thermodynamics for discontinuous maps and potentials

Abstract :

 The basic result of thermodynamic formalism is the variational principle, which is usually stated under the hypotheses that the space is compact, the map is continuous, and the potential is continuous.  Many natural examples violate one or more of these conditions, but still satisfy some version of the variational principle.  In the literature this usually relies on some underlying structure of the system, such as a symbolic representation.  I will present a version of the variational principle that holds quite generally.  This has applications to studying uniqueness of equilibrium states without the assumptions of continuity and compactness.

 
 
 
 
Name
Daniel Coronel
Institution
Universidad Andres Bello
Title:

Phase transitions in the quadratic family.

Abstract :

We give the first examples of transitive quadratic maps having a phase transition after the first zero of the geometric (real and complex) pressure function. More precisely, we show two examples of phase transitions, one of first order and the other of higher order. In each example the quadratic map has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.
This is a joint work with Juan Rivera-Letelier

 
 
 
 
Name
Katrin Gelfert
Institution

Instituto de Matematica Universidade Federal do Rio de Janeiro

https://mail.google.com/mail/u/0/images/cleardot.gifTitle:

Non-hyperbolic structures of geodesic flows on compact rank-1 surfaces

Abstract :

I will talk about some properties of geodesic flows of a compact  succesfully applied to primary examples in conformal dynamics. surface in the case that the flow is non-uniformly hyperbolic. I will discuss some approaches that allow us to study orbits that lack uniformly hyperbolic behavior. Such approaches can be used, for example to study level sets of Lyapunov regular points with equal exponent. If the flow is not Anosov, then the set of points with Lyapunov exponents close to zero can be quite large or small (when measured e.g. in terms of fractal dimension or topological entropy). This is investigated by means of the thermodynamic formalism for sub systems that are uniformly hyperbolic. Such a scheme can be succesfully applied to primary examples in conformal dynamics.

 
 
 
 
Name
Oliver Jenkinson
Institution
Queen Mary,
Title:

Ergodic optimization, ergodic dominance, and thermodynamic formalism

Abstract :

In this mini-course we will compare and contrast ergodic optimization (i.e. the study of optimizing ergodic averages) with aspects of thermodynamic formalism, and explore the related notion of ergodic dominance (i.e. equipping the set of invariant probability measures with a "stochastic dominance" partial order).

 
 
 
 
Name
Thomas Michael Jordan
Institution
University of Bristol
Title:

Multifractal analysis for quotients of Birkhoff sums for countable Markov maps

Abstract :

We consider the multifractal analysis of quotients of Birkhoff> averages for countable Markov maps. We prove a variational principle for  the Hausdorff dimension of the level sets. Under certain assumptions we  are able to show that the spectrum varies analytically in parts of its  domain. We apply our results to show that the Birkhoff spectrum for the  Manneville-Pomeau map can be discontinuous, showing the considerable  differences between this setting and the uniformly hyperbolic setting.  This is joint work with Godofredo iommi.

 
 
 
 
 
Name
Dominik Kwietniak
Institution
Jagiellonian University in Krakow
Title:

When the simplex of invariant measures is Poulsen?

Abstract :

I will talk about the set of conditions which guarantee that
the simplex of invariant measures of a dynamical system (X,T) is the
Poulsen simplex. These conditions allow also to prove that every
measure has a generic point. They are fulfilled for example by all
beta shifts and S-gap shifts. The similar albeit non-equivalent
approach was recently presented by Climenhaga and Thompson. If time
permits, then I will discuss the construction of some examples of
entropy functions on a simplex of invariant measures with curious
properties (for example, a systems with a dense set of ergodic
measures but with only one ergodic measure of positive entropy).
This
relies on Downarowicz entropy structure machinery.

 
 
 
 
Name
Tamara Kucherenko
Institution
CCNY
Title:

 "Geometry and Entropy of Generalized Rotation Sets".

Abstract :

 For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized  rotation set $R(Phi)$. Here $Phi=(phi_1,...,phi_m)$ is a $m$- dimensional continuous potential and $R(Phi)$ is the set of all $mu$-integrals of $Phi$ and $mu$ runs over all $f$- invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of $R^m$. We show that in the case of subshifts of finite type every compact and convex set in $R^m$ can be realized as a rotation set of a continuous potential. We also discuss the relation between $R(Phi)$ and the set of all statistical limits $R_{Pt}(Phi)$. We illustrate with examples that in general these sets differ but are able to provide conditions that guarantee $R(Phi)= R_{Pt}(Phi)$. Finally, we will look at certain properties of the entropy function $wmapsto H(w), win R(Phi)$. This is a joint work with Christian Wolf

 
 
 
 
Name
Artur Lopes
Institution
Universidade Federal do Rio Grande do Sul
Title:

Entropy and Variational Principle for  one-dimensional Lattice Systems with a general a-priori measure, jooint work with J. Mengue, J. Mohr and R. Souza

Abstract :

We generalize several results of the classical theory of Thermodynamic Formalism  by considering a    compact metric space M as the state space.We analyze the  shift acting on M^N and consider a  general a-priori measure for defining the Ruelle operator.We study potentials Lipchitz A which can depend on the infinite set of coordinates in M^N.We define entropy and by its very nature it is always a nonpositive number. The concepts of entropy and transfer operator are linked. We analyze the Pressure problem and we show the existence of Gibbs and equilibrium probabilties.As a particular case we analyze the problem when M=S^1 ( the unitary circle )  and the a-priori measure is Lebesgue dx, the infinite product of dx on (S^1)^N We show that the eigenfunction is differentiable when the potential A is differentiable.We also show that the study of Gibbs states for the  Bernoulli  space with countable infinite symbols can be analyzed by adapting this formalism.

 
 
 
 
Name
 Ian Morris
Institution
University of Surrey
Title:

Running times of GCD algorithms and the pressure of random dynamical systems which expand on average

Abstract :

The estimation of the average number of division steps required to complete the operation of the Euclidean algorithm was first investigated by Heilbronn and Dixon in the early 1970s using explicit combinatorial arguments. In the 1990s, researchers such as Douglas Hensley and Brigitte Vallee noted that the asymptotics of the average number of steps could be understood using the perturbation theory of the Ruelle operator associated to the continued fraction transformation together with a Tauberian argument, which led to extensions of Heilbronn and Dixon´s results in two directions: firstly the proof of a central limit theorem for the asymptotic distribution of the number of division steps, and secondly the application of the method to a broader class of GCD algorithms. We describe progress towards some open problems in this area such as the asymptotic behaviour of the binary and 2-adic Euclidean algorithms, which are described by the spectra of Ruelle operators corresponding to certain IID random dynamical systems.

 
 
 
 
Name
Andres Navas
Institution
USACH
Title:

"Invariant distributions for circle diffeomorphisms and a equidistribution theorem à la Weyl / Herman".

Abstract :

We show that circle diffeomorphisms in the Denjoy class having irrational rotation number carry no invariant 1-distribution other than finite invariant measures. As a corollary, we obtain an equidistribution theorem for smooth potentials. Using this, we retrieve in a conceptual way a theorem of M. Herman concerning the C^1 convergence to the identity of certain iterates of C^2 circle diffeomorphisms. (This is joint work with M. Triestino.)

 
 
 
 
Name
Krerley Oliveira
Institution
Universidade Federal de Alagoas
Title:

Uniqueness of Non-lacunary Gibbs Measures: the non-uniformly expanding case

Abstract :

 General conditions  to obtain existence and uniqueness of equilibrium states and Gibbs measures are not known beyond uniform expan-ding/hyperbolic setting. In this talk, we discuss a point of view to treat maps such that equilibrium states are hyperbolic measures. One could ask if under reasonable conditions,  among hyperbolic measures with the same number of positive and negative Lyapunov exponents, there exists an unique equilibrium measure?  We give an affirmative answer for the previous question for the case of measures with only positive Lyapunov exponents. For a continuous map f and potential φ with total pressure of the set of points with innitely many hyperbolic times admits at most one equilibrium measure, provided that f has some expanding conformal measure. Moreover, to prove this, we develop an non-uniform version of the Ruelle-Perron-Frobenius Theorem and we show that if, in addition, f is transitive and φ has a conformal measure with innitely many hyperbolic times, then there exists at most one unique equilibrium measure, which is absolutely continuous with respect to a non-lacunary Gibbs measure. Finally, we obtain the existence of this equilibrium measure, assuming that the pressure of the points with innitely many times is bigger than the pressure of its complement.

 
 
 
 
Name
Mark Pollicott
Institution
University of Warwick
Title:

Analyticity of the Hausdorff Dimension of Horseshoes

Abstract :

We will describe a simple method for showing the Hausdorff Dimension of a two dimensional horseshoe depends analytically on the diffeomorphism.  This result has applications to the theory of Discrete Schrodinger Operators.

 
 
 
 
Name
Michael Rams
Institution
IMPAN
Title:

Multifractal formalism for infinite systems

Abstract :

 I plan to present an introduction to the multifractal theory and then proceed to the special case of infinite iterated function systems. My main topic will be the Birkhoff spectrum.

 
 
 
 
Name
Jörg Schmeling
Institution
University of Lund
Title:

Multifractal analysis of some multiple ergodic average

Abstract :

 Abstract. Let (X; T) be a topological dynamical system where T is a continuous map on a compact metric space X. Furstenberg had initiated the study of the multiple ergodic average:   

  

1/n sum_k=1}^n   f_1(T^k x)f_2(T^2k x)....f_s(T^sk x)                                     (1)

  

 

where f1; … ; fs are s continuous functions on X with s geq 2 when he proved the existence of arithmetic sequences of arbitrary length amongst sets of integers with positive density. Later on, the research

of such a kind of average has attributed a lot of attentions. We study the multiple ergodic averages

  

 1/n sum_k=1}^n φ(x_k; x_kq; .................     ; x_kq^ℓ-1 )

  

 

on the symbolic space Sigma_m = (0; 1; …. ;m-1)^N where m geq 2; l geq 2; qgeq 2 are integers. First we will give a solution to the problem of multifractal analysis of the limit of the multiple ergodic averages for Bernoulli functions. We will continue by considering the invariant part of the multifractal level sets, i.e. we will study the maximal dimension of an invariant or multiple mixing measure supported on these level sets.

Here many new interesting phenomena occur. In general there will be no invariant measure with the same dimension as the level sets. Moreover the invariant and the mixing spectra di er. On the other

hand we will point on some connections to probability theory (von Mise statistics), ergodic optimization of multiple integrals and also indicate some new phase transition phenomena. This is joint work with Ai-Hua FAN, and Meng WU.

 
 
 
 
Name
John Richard Sharp
Institution
University of Warwick
Title:

Higher Teichmüller theory, generalised lengths and thermodynamic formalism

Abstract :

Recently there has been considerable interest in so-called higher Teichmüller theory, i.e. the study of certain representations of the fundamental groups of surfaces in PSL(n,R) for n geq 3. It turns out that there are strong parallels with the classical case (n=2), in particular analogues of the Prime Geodesic Theorem describe the asymptotics of generalised lengths which arise in this setting. It turns out that these quantities are may be studied via flows related to the classical geodesic flow over the original surface using the thermodynamic ergodic theory familiar there. This is joint work with Mark Pollicott.

 
 
 
 
Name
Pablo Shmerkin
Institution
University of Surrey
Title:

 "Continuity of subadditive pressure for matrix products".

Abstract :

For conformal dynamical systems, the Hausdorff dimension of invariant sets is often expressed as the zero of certain pressure equation, which is easily seen to be continuous in the defining dynamics. The situation is dramatically more complicated in the non-conformal setting where, nevertheless, a subadditive pressure equation involving singular values of a matrix cocycle plays a crucial role. We prove that in the locally constant case, this geometrical subadditive pressure is continuous as a function of the defining cocycle, answering a folklore question in the fractal geometry community. The proof uses the variational principle for subadditive potentials in a crucial way.This is joint work with De-Jun Feng (Chinese University of Hong Kong).

 
 
 
 
Name
Bartlomiej Skorulski
Institution
Universidad Catolica del Norte
Title:

Regularity and irregularity of fiber dimension of non-autonomous dynamical systems

Abstract :

I will talk about non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. This talk is based on joint work woth Volker Mayer and Mariusz Urbanski where we provided a necessary and sufficient condition for holomorphic stability which leads to Holder continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the topology on the parameter space. On the other hand we showed that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space.

 
 
 
 
Name
Manuel Stadlbauer
Institution
Universidade Federal da Bahia.
Title:

On a Perron-Frobenius theorem for transient group extensions

Abstract :

In this talk, we study thermodynamic formalism of group extensions and its relation to boundary theory for Markov chains. In the context of shift spaces, the skew product T :X x G ->X x G; (x;g) ->(Θ(x);gΨ(x))     is referred to as a group extension, where Θ : X -> X is a topologically mixing, countable Markov shift, G a discrete group and Ψ :X -> G a map which is measurable with respect to the canonical partition. By considering a Holder continuous potential Φ on X and its associated Gibbs measure μ for Θ, we obtain an object which might seen as a random walk on G with stationary increments distributed according to μ o Ψ·¹ Through construction of a family ofs-nite conformal measures, it is then possible to obtain a σ-nite Ruelle-Perron-Frobenius theorem. That is, if X satises the big images and big preimages property, then there exists a s-nite conformal measure and a Holder continuous eigenfunction. However, note that an application of a theorem ¨ of Zimmer gives that the group extension has to be totally dissipative if G is nonamenable. Furthermore, the above RPF-theorem provides a rich family of harmonic functions which, in particular, has applications to the boundary theory of the random walk with stationary increments.

 
 
 
 
Name
Daniel Thompson
Institution
The Ohio State University
Title:

Equilibrium states and large deviations for systems with non-uniform structure

Abstract :

I will give an overview, and report on recent progress, for a long-term project joint with Vaughn Climenhaga concerning measures of maximal entropy and equilibrium states for a large class of dynamical systems with a ´non-uniform orbit structure´, including piecewise continuous and parabolic interval maps. A recent advance in this project, which is joint work with Kenichiro Yamamoto, is the establishment of the level-2 large deviations principle in this setting.

 
 
 
 
Name
Michael John Todd
Institution
St Andrews University,
Title:

Dynamical systems with holes: escape rates, equilibrium states and dimension

Abstract :

Given a smooth interval map with one or more critical points, we consider the dynamical behaviour of the system when a hole is punctured in the system, allowing mass to escape. Work by Bruin, Demers and Melbourne studied the escape of Lebesgue measure, obtaining a conditionally invariant measure absolutely continuous w.r.t. Lebesgue, as well as a related measure on the survivor set (the set of points which never escape).  This was related to the study of equilibrium states. In this talk, motivated by Bowen´s formula for the dimension of dynamically defined sets, I´ll consider escape w.r.t. other natural measures and thus give an expression for the Hausdorff dimension of the survivor set.  This is joint work with M. Demers.

 
 
 
 
 
 
 
 
Name
Mariusz Urbanski
Institution
University of North Texas
Title:

"Fine Inducing and Equilibrium Measures for Rational Functions of the Riemann Sphere"

Abstract :

 Let $f:mathbb{C} o mathbb{C}$ be an arbitrary holomorphic endomorphism of degree larger than $1$ of the Riemann sphere $mathbb{C}$.. Denote by  $J(f)$ its Julia set. Let $varphi:J(f) omathbb{R}$ be a H"older continuous function whose topological pressure exceeds its supremum.  It is known that then there exists a unique equilibrium measure $mu_varphi$ for this potential. I will discuss a special inducing scheme with fine recurrence properties. This construction allows us to prove  three results. Dimension rigidity, i.e. a characterization of all maps and potentials for which $HD(mu_varphi)=HD(J(f))$. Real analyticity of topological pressure $P(tvarphi)$ as a function of $t$. Exponential decay of correlations, and, as its consequence, the Central Limit Theorem and the Law of Iterated Logarithm for H"older continuous observables. Finally, the Law of Iterated Logarithm for all linear combinations of H"older continuous observables and the function $log|f´|$. Geometric consequences of the  Law of Iterated Logarithm lead to comparison of equilibrium states with  appropriately  generalized Hausdorff measures on the Julia set $J(f)$. 

 
 
 
 
Name
Paulo Varandas
Institution
Universidade Federal da Bahia.
Title:

Weak specification properties and large deviations for non-additive potentials

Abstract :

In this talk we address the existence of large deviation principles in the case of non-additive potentials. We obtain some large deviation upper and lower bounds for the measure of deviation sets associated to asymptotically additive and sub-additive sequences of potentials under some weak specification assumption for the dynamical system. In particular we present a large deviation principle in the case of uniformly hyperbolic dynamical systems and also some applications to the speed of convergence to Lyapunov exponents.(joint work with Yun Zhao)

 
 
 
 
Name
Christian Wolf
Institution
CCNY
Title:

Relative topological pressure, rotation sets and equilibrium states

Abstract :

We introduce a notion of relative topological pressure and establish for several classes of systems and potentials a variational principle. We also provide examples showing that in general a relative variational principle does not hold. Finally, we discuss the notion of relative equilibrium states and show that even in the case subshifts of finite type and H"older continuous potentials these equilibrium states are in general not unique.