Thermodynamic Formalism and applications
July 812, 2013
PUC. Santiago, Chile
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Thermodynamic Formalism and applications is a meeting that will focus on recent developments of this theory.
We expect to gather researchers on the subject to create a working atmosphere and to facilitate scientific interactions.
The program will consist of minicourses and talks.
It will be held from the 8th of July to the 12th of July 2013, at the Facultad de Matemáticas of Pontificia Universidad Católica de Chile in Santiago, Chile.
Organizing Committee
Godofredo Iommi (PUC)
Mario Ponce (PUC).
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 nonrecurrent critical point, so it is nonuniformly hyperbolic in a strong sense. 





Name

Katrin Gelfert

Institution

Instituto de Matematica Universidade Federal do Rio de Janeiro 
Title: 
Nonhyperbolic structures of geodesic flows on compact rank1 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 nonuniformly 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 minicourse 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 MannevillePomeau 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 





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 onedimensional Lattice Systems with a general apriori 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 apriori 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 apriori 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 2adic 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 1distribution 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 Nonlacunary Gibbs Measures: the nonuniformly expanding case 

Abstract :

General conditions to obtain existence and uniqueness of equilibrium states and Gibbs measures are not known beyond uniform expanding/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 inﬁnitely many hyperbolic times admits at most one equilibrium measure, provided that f has some expanding conformal measure. Moreover, to prove this, we develop an nonuniform version of the RuellePerronFrobenius Theorem and we show that if, in addition, f is transitive and φ has a conformal measure with inﬁnitely many hyperbolic times, then there exists at most one unique equilibrium measure, which is absolutely continuous with respect to a nonlacunary Gibbs measure. Finally, we obtain the existence of this equilibrium measure, assuming that the pressure of the points with inﬁnitely 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^ℓ1 )
on the symbolic space Sigma_m = (0; 1; …. ;m1)^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 AiHua 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 socalled 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 nonconformal 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 DeJun Feng (Chinese University of Hong Kong). 





Name

Bartlomiej Skorulski

Institution

Universidad Catolica del Norte

Title:

Regularity and irregularity of fiber dimension of nonautonomous dynamical systems 

Abstract :

I will talk about nonautonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of nonautonomous 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 nonautonomous 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 PerronFrobenius 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 ofsﬁnite conformal measures, it is then possible to obtain a σ–ﬁnite RuellePerronFrobenius theorem. That is, if X satisﬁes the big images and big preimages property, then there exists a sﬁnite 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 RPFtheorem 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 nonuniform structure 

Abstract :

I will give an overview, and report on recent progress, for a longterm project joint with Vaughn Climenhaga concerning measures of maximal entropy and equilibrium states for a large class of dynamical systems with a ´nonuniform 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 level2 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 $logf´$. 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 nonadditive potentials 

Abstract :

In this talk we address the existence of large deviation principles in the case of nonadditive potentials. We obtain some large deviation upper and lower bounds for the measure of deviation sets associated to asymptotically additive and subadditive 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. 
1. Mahsa Allahbakhshi (Santiago)
2. Alexis Bailler (Santiago)
3. Rodrigo Bissacot (Sao Paulo)
4. Jairo Bochi (Rio de Janeiro)
5. Jose Bravo (Sao Carlos)
6. Vaughn Climenhaga (Houston)
7. Daniel Coronel (Santiago)
8. Maria Isabel Cortez (Santiago)
9. Katrin Gelfert (Rio de Janeiro)
10. Irene Inoquio Renteria (Valdivia)
11. Oliver Jenkinson (London)
12. Thomas Jordan (Bristol)
13. Jan Kiwi (Santiago)
14. Tamara Kucherenko (New York)
15. Dominik Kwietniak (Krakow)
16. Huaibin Li (Santiago)
17. Amanda de Lima (Sao Paulo)
18. Artur Lopes (Porto Alegre)
19. Ian Morris (Surrey)
20. Andres Navas (Santiago)
21. Krerley Oliveira (Meceio)
22. Mark Pollicott (Warwick)
23. Luciano Prudente (Rio de Janeiro)
24. Michal Rams (Warsaw)
25. Jörg Schmeling (Lund)
26. Richard Sharp (Warwick)
27. Michael Schraudner (Santiago)
28. Pablo Shmerkin (Surrey)
29. Carlos Siqueira (Sao Paulo)
30. Bartlomiej Skorulski (Antofagasta)
31. Manuel Stadlbauer (Bahia)
32. Mahdi Teymuri Garakani (Santiago)
33. Dan Thompson (Ohio)
34. Mike Todd (St. Andrews)
35. Mariusz Urbanski (Denton)
36. Paulo Varandas (Bahia)
37. Anibal Velozo (Santiago)
38. Christian Wolf (New York)
39. Yiwei Zhang (Santiago)
MiniCourses
1. Oliver Jenkinson (Queen Mary).
Title: Ergodic optimization, ergodic dominance, and thermodynamic formalism
Abstract: In this minicourse 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).
2. Michal Rams (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 systmes. My main topic will be the Birkhoff spectrum.
1. HOTEL NERUDA EXPRESS.
The following are the contact details of the hotel:
There are 3 ways to go from Santiago Airport to the Hotel.
From the Hotel Neruda Express to University.
2. HOTEL GEN.
The following are the contact details of the hotel:
FROM THE AIRPORT TO HOTEL GEN
From Hotel Gen to University
Depending on the time of the day (peak hours: from 7am to 9:30am & from 5:30pm to 9pm) some of the trains stop only every other station. They have a RED or a green light to indicate the stations they stop at. The San Joaquin metro station is a RED station.
In order to have dinner next to the hotel there are several places you can go. Barrio Lastarria has several restaurants and bars. In order to get there you have to walk 4 blocks north on Avenida Portugal until Alameda Bernardo O´ Higgins and the cross that street. Another option is to go to Barrio Italia which also has a good variety of restaurants to choose from. In order to get there you have to walk one block down on Avenida Portugal, then about 3 blocks east on Avenida Santa Isabel until Avenida Italia or parque Bustamante.
The conference will be held at the Audotorio Niroslav Bralic at the Facultad de Matematica of Pontiﬁcia Universidad Catolica de Chile. This is located at the Campus San Joaquin of the University. Note that there are several campus spread over Santiago.
(a) Address: Avenida Vicuna Mackenna 4860, Santiago, Chile.
A map of the campus with the location of the conference room can be found at
www.sistemasdinamicos.cl/conferences/termo2013/webcontent_182.html
In order to take a bus in Santiago you need a card, the so called «BIP» card. It can be purchased at any metro station at a cost of 2000 CLP. You need to charge it with certain amount of money and then use it to pay your fare. The cost of the bus ticket is more or less: 600 CLP.
You might need to travel with health insurance because the public system is not very good and the private one is expensive.
For any informartion regarding the conference or if you want to register please send an email to Godofredo Iommi to the following address:
giommi (at) mat (dot) puc (dot) cl
Center of Dynamical Systems and Related Fields (ACT project 1103 PIACONICYT).
Pontificia Universidad Católica de Chile (VRI y Facultad de Matemáticas).
On Wednesday 10th of July at 20:30 there will be a conference dinner at the restaurant «El Mesón Nerudiano», which is located at Dominica 35, Barrio Bellavista. There will be minibuses driving people from the hotels to the restaurant and back again.