Thermodynamic Formalism: Non-additive Aspects and Related Topics

14.05.2023 - 19.05.2023 | Będlewo


See schedule and program.


Anton Gorodetski | Non-stationary version of Furstenberg Theorem on random matrix products

We provide a non-stationary analog of the Furstenberg Theorem on random matrix products (that can be considered as a matrix version of the law of large numbers). Namely, under a suitable genericity conditions the sequence of norms of random products of independent but not necessarily identically distributed SL(d,ℝ) matrices grow exponentially fast, and there exists a non-random sequence that almost surely describes asymptotical behavior of that sequence. These are the results of a joint project with Victor Kleptsyn.

Antti Käenmäki | Finer geometry of planar self-affine sets

For a planar self-affine set satisfying the strong separation condition, it has been recently proved that under mild assumptions the Hausdorff dimension equals the affinity dimension. In this article, we continue this line of research, and our objective is to acquire more refined geometric information in this setting. In a large class of such non-carpet planar self-affine sets, we characterize the Ahlfors regularity, study Marstrand's projection theorem, and show that Marstrand's slicing theorem cannot be extended to all slices. The talk is based on recent work with Balázs Bárány and Han Yu.

Balázs Bárány | Multifractal analysis of weighted Birkhoff averages

In this talk, we will study the spectrum of weighted Birkhoff averages of continuous potentials over aperiodic and irreducible subshifts of finite type. We show that the topological entropy spectrum of weighted Birkhoff averages with monotone decreasing weights equals the spectrum of the usual Birkhoff averages. Moreover, we show that the packing spectrum of such weighted Birkhoff averages equals either the packing spectrum of the usual Birkhoff averages or it is the constant value of the dimension of the whole space depending on some properties of the weights. This talk is based on the joint work with Michał Rams and Ruxi Shi.

De-Jun Feng | Typical self-affine sets with non-empty interior

A self-affine set is the attractor of an affine iterated function system.  In his celebrated work, Falconer (1988) determined the dimension of typical self-affine sets. Later, Jordan, Pollicott and Simon (2007) gave a sufficient condition for a typical self-affine set to have positive Lebesgue measure.   In this talk, I will present some sufficient conditions for a typical self-affine set to have a non-empty interior. This is based on joint work with Zhou Feng.  

Henna Koivusalo | Shrinking targets and iterated function systems, a thermodynamic take

The classical shrinking target problem concerns the following set-up: Given a dynamical system (T, X) and a sequence of targets (Bn) of X, we investigate the size of the set of points x of X for which Tn(x) hits the target Bn for infinitely many n. In this talk, I will discuss shrinking target problems in the context of iterated function systems, where `size' is studied both from the perspective of dimension and measure. The dimension results have a natural interpretation as a pressure formula from the perspective of thermodynamic formalism. I will give an overview of the topic, aiming also to cover some of my own results, joint with various subsets of Simon Baker, Thomas Jordan, Lingmin Liao, and Michał Rams.

Ian Morris | A variational principle relating self-affine measures and self-affine sets

A breakthrough result of ‪Bárány‬, Rapaport, and Hochman published in 2019 showed that if a two-dimensional affine iterated function system is strongly irreducible and satisfies the strong open set condition, then the Hausdorff dimension of its attractor is equal to a value defined by Falconer in 1988. Their result applied a Ledrappier Young formula established by Bárány and Käenmäki and a variational principle due to Morris and Shmerkin in combination with deep new results on projections of self-affine measures. This Ledrappier-Young formula has since been extended to higher dimensions by Feng, and projections of self-affine measures in higher dimensions are currently being studied by Rapaport. In this talk, I will describe an extension of the variational principle to higher dimensions. In combination with a recent preprint of Rapaport this implies that the above theorem of Bárány‬, Rapaport and Hochman is also valid in dimension three. This is joint work with Çağrı Sert.

Joanna Kułaga-Przymus | Thermodynamic formalism for B-free systems

Given B ⊆ N, we consider the corresponding set F_B of B-free integers, i.e. n ∈ F_B iff no b ∈ B divides n. We define X_η - the B-free subshift - as the smallest subshift containing η := 1_{F_B} ∈{0, 1}^Z. Such systems are interesting both from the dynamical and number theoretical viewpoint and can manifest various types of behavior. I will concentrate on the entropy and its generalization – topological pressure, with some applications in combinatorics/number theory. Talk will be based on a joint work with Michał Lemanczyk and Michał Rams.

Julien Barral | Sparse sampling and dilation operations on a Gibbs weighted tree, and multifractal formalism

Starting from a Gibbs capacity, we build a new random capacity by applying two simple operations, the first one introducing some redundancy and the second one performing a random sampling.  Depending on the values of the two parameters ruling the redundancy and the sampling, the new capacity has very different multifractal behaviors. In particular,  the multifractal spectrum of the capacity may contain two to four phase transitions, and the multifractal formalism may hold only on a strict subset (sometimes, reduced to a single point) of the spectrum's domain. This is joint work with S. Seuret.

Karoly Simon | Continuous piecewise linear iterated function systems on the line

In the last four decades, considerable attention has been given to studying self-similar Iterated Function Systems (IFS). On the line, a self-similar IFS consists of finitely many contracting similarity mappings. In this talk, we consider more general IFSs on the line. Namely, when the IFS consists of finitely many continuous piecewise linear functions such that each slope is different from zero and less than one in absolute value. These are the Continuous Piecewise Linear IFSs (CPLIFS). After a review of our earlier result related to the dimension of the attractor of CPLIFSs, we introduce some conditions under which the attractor of a CPLIFS equals the attractor of a graph-directed self-similar attractor. Moreover, we give conditions which imply that, in some sense, typically, the attractor of a CPLIFS has a positive Lebesque measure. The new results presented are joint with  Daniel Prokaj and Peter Raith.

Kenneth Falconer | Fractals, multifractals and subadditive thermodynamic formalism

The notion of extending the thermodynamic formalism to a nonadditive setting goes back to the 1980s, at least partly motivated by the desire to apply it to dimension theory of self-affine or non-conformal recursively defined fractals.

The talk will discuss some of the questions that have been pondered since then, on which remarkable progress has been made but where many difficult problems remain open.

Laura Breitkopf | Distribution of Stern–Brocot sequences generalized to Hecke triangle groups

The Stern-Brocot sequence, known from number theory, can be studied within the framework of dynamical systems. To do so, the sequence is constructed from a continuous transformation, the Farey mapping. We construct new generalized Stern-Brocot sequences through a generalization of the Farey mapping by means of Hecke triangle groups. In this talk, we introduce these sequences and discuss our recent contributions to the study of their properties, which involves methods from infinite ergodic theory. In particular, we reproduce a distribution result of Keßeböhmer and Stratmann for the classical Stern-Brocot sequence and extend it to our generalized sequences. This is joint work with Anke Pohl and Marc Keßeböhmer.

Mark Pollicott | Upper bounds on the Hausdorff dimension of the Rauzy gasket

The Rauzy gasket is the limit set of three non-expanding maps of the two-dimensional simplex into itself. It plays a role, for example,  in the theory of interval exchange transformations. Since the maps are not strict contractions nor conformal this makes estimates on its dimension delicate. Avila-Hubert-Skripchenko showed that the Hausdorff dimension is strictly smaller than 2. Fougeron improved on this upper bound.  We show that the Hausdorff dimension is bounded above by 1.7415.  This is joint work with Ben Sewell.

Mike Todd | Cover times in dynamical systems

What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)?  This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem.  I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics and transfer operators as well as permutations and inducing.

Noé Cuneo | Large deviations of return times and related entropic estimators on shift spaces

The study of return times in dynamical systems has a long history, and their role as entropy estimators is well established. While considerable attention has been devoted to the corresponding Law of Large Numbers, Central Limit Theorem, cumulant-generating function and multifractal spectrum in the literature, surprisingly little was known about their large deviations. In fact, only local versions of the Large Deviation Principle (LDP) were obtained, and then only in the case of equilibrium measures for Bowen-regular potentials on shift spaces. I will talk about an upcoming paper with Renaud Raquépas, where we prove that return times satisfy the full LDP for a large class of invariant measures on shift spaces. The measures are subject to some decoupling conditions, which imply no form of mixing nor ergodicity, and which allow to go even beyond the scope of thermodynamic formalism. As we will understand through simple examples, the rate function is nonconvex in general.

Polina Vytnova | Discrepancy between dimensions of self-similar measures

We show that the Hausdorff, the correlation and the Frostman dimensions of Bernoulli convolution measures corresponding to Pisot parameter values can be written in terms of the joint  spectral radius and Lyapunov exponent of a family of finite dimensional matrices. As a corollary we show that these dimensions do not agree. Furthermore, it turns out that these dimensions also admit a “thermodynamic” interpretation in terms of a certain “free energy” convex function.

Ronnie Pavlov | Uniqueness of measure of maximal entropy for subshifts with few forbidden words

It's well-known that any irreducible shift of finite type has unique measure of maximal entropy (the so-called Parry measure). Phrased another way, this means that forbidding a finite number of words from a full shift (and irreducibility) is enough to guarantee unique MME. For an infinite list of forbidden words, it's natural to quantify by the number F(n) of forbidden words of each length. In recent work, I showed that when F(n) grows slowly enough, the subshift has a unique MME which has the K-property. The proof involves a sort of measure-theoretic version of the specification property, which may be of independent interest and which I will outline if time permits.

Sascha Troscheit | Diophantine approximation and coverings on random self-similar and self-affine sets

Khintchine’s theorem is an important result in number theory which links the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. This behaviour has been observed for deterministic (fractal) sets and serves as the inspiration to investigate the random settings. Introducing randomisation into the problem makes some parts more tractable, while posing separate new challenges. In this talk, I will present joint work with Simon Baker where we provide sufficient conditions for a large class of stochastically self-similar and self-affine attractors to have positive Lebesgue measure.

Stephane Seuret | Measures, annuli and dimensions

Given a Radon probability measure m, we are interested in those points x around which the measure is concentrated infinitely many times on thin annuli centered at x. Such questions related to spherical averages arises in several situations: existence of "bad points"  for the Poincaré recurrence theorem,  first return times to shrinking balls under iteration generated by a weakly Markov dynamical system, Kakeya-type problems and Falconer’s distance set conjecture. Depending on the lower and upper dimension of m,  the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of m-measure 0 or 1. This is a joint work with Z. Buczolich. 

Vaughn Climenhaga | Specification for non-compact systems

I will describe joint work with Dan Thompson and Tianyu Wang in which we formulate a version of Bowen’s specification property suitable for non-compact systems and use it to prove uniqueness of equilibrium states under a strong positive recurrence condition on the potential function. Our results have applications to geodesic flows on non-compact manifolds with negative curvature.

Victor Kleptsyn | Hölder regularity of stationary measures

One of the main tools of the theory of dynamical systems are the invariant measures; for random dynamical systems, their role is taken by stationary measures, measures that are equal to the average of their images. In a recent work with A. Gorodetski and G. Monakov, we show that these measures  almost always (under extremely mild assumptions) satisfy the Hölder regularity property:  the measure of any ball is bounded by (a constant times) some positive power of its radius.

Yuri Lima | Symbolic dynamics for large non-uniformly hyperbolic sets of three-dimensional flows

In a joint work with Jérôme Buzzi and Sylvain Crovisier, we construct symbolic dynamics for three-dimensional flows with positive speed. More precisely, for each χ > 0, we code a set of full measure for every invariant probability measure which is χ–hyperbolic. These include all ergodic measures with entropy bigger than χ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of [−χ, χ]. This contrasts with previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code χ–hyperbolic measures in homoclinic classes of measures by irreducible countable Markov shifts.

Abstracts of shorter contributions:

Adam Śpiewak | Absolute continuity of self-similar measures on the plane

We consider iterated function systems consisting of contracting similarities on the complex plane and prove that for almost every choice of the contraction parameters in the super-critical region (i.e. with similarity dimension greater than 2) the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest. This is joint work with Boris Solomyak.

Alex Rutar | Random substitutions and multifractal analysis

A (deterministic) substitution consists of a finite alphabet along with a set of transformation rules. Random substitutions are a generalization which allow multiple transformation rules for each letter, along with associated probabilities. Associated with a substitution is a shift-invariant ergodic frequency measure, which quantifies the relative occurrence of finite words as subwords of the substitution. Frequency measures are an interesting class of invariant measures which witness a form of self-similarity, while exhibiting complex overlapping phenomena.  In this talk, I will provide a general introduction to random substitutions as well as their dimensional properties via the Lq-spectrum. I will also discuss a particular class of random substitutions for which the Lq-spectrum is analytic on and for which we can prove a variational principle (which in particular implies the multifractal formalism). Perhaps interestingly, however, the measures themselves are not in general Gibbs. This is based on joint work with Andrew Mitchell (Birmingham).

Amlan Banaji | Dimensions of continued fraction sets

Sets of numbers which have real or complex continued fraction expansions with entries restricted to a given countable set can be viewed as limit sets of infinite conformal iterated function systems. In this setting, different notions of fractal dimension, such as Hausdorff, box and Assouad dimension, can take different values. We will describe what is known about these dimensions and two other families of dimensions, namely the intermediate dimensions and Assouad spectrum, which provide more refined global and local information respectively about the sets. This is based on joint work with Jonathan Fraser.

Andrzej Biś | Entropy functions for semigroup actions

Using methods from Convex Analysis, for each generalized pressure function we define an upper semi-continuous affine entropy-like map, establish an abstract variational principle for both countably and finitely additive measures and prove that equilibrium states always exist. We study the thermodynamic formalism of continuous actions of semigroups generated by continuous self-maps or homeomorphisms of a compact metric space X. This setting comprises finitely generated semigroups, countable sofic groups and uncountable groups endowed with a reference probability measure. For each topological pressure operator associated to these actions we provide both an affine, upper semi-continuous entropy-like map, whose domain is the set of Borel probability measures on X, and a variational principle whose maximum is always attained. The talk is based on joint work with Maria Carvalho, Miguel Mendes and Paulo Varandas.

Carlos Vasquez | Intermingled phenomena for Kan diffeomorphisms

In 1994, Ittai Kan provided the first examples of maps with intermingled basins. The maps correspond to skew products on the cylinder, with a uniformly expanding map in the base and having two physical measures supported on the boundary of the cylinder. Two physical measures are intermingled if, for every open set U in the cylinder, each basin has a positive Lebesgue measure on U. Note that the Lebesgue measure on the cylinder plays the role of reference measure in the definition of physical measure and intermingles phenomenon. Throughout this talk, we will discuss the intermingled phenomenon when the reference measures are other than Lebesgue. This is a joint work with Lorenzo Díaz (PUC-Rio); Katrin Gelfert (UFRJ) and Bárbara Nuñe-Madariaga from PUCV.

Daniel Prokaj | On the attractor of piecewise linear iterated function systems

Consider an iterated function system on the line, which consists of continuous, piecewise linear functions. Previously, we proved that typically the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained from the most natural covering system, which we call the natural dimension of the IFS. In this talk, we are going to show that for Lebesgue typical parameters, the Lebesgue measure of the attractor is strictly positive if the natural dimension of the system is bigger than one, and all the contraction ratios are positive. This talk is based on a joint article with Károly Simon. (arXiv:2212.09378)

Hasan Akin | On the asymptotic behavior of thermodynamic quantities of Ising model with mixed spin (s,(2t−1)/2) on a Cayley tree by self-similarity method

Spin systems, including the hard-core, Ising, and Potts models, constitute a broad spectrum of probabilistic models studied in applied probability, statistical physics, artificial intelligence, and other fields. In statistical mechanics, there are essentially two rigorous methods for investigating the thermodynamic quantities connected to spin systems on the Cayley tree (CT): self-similarity and Kolmogorov consistency theorem. Using the Kolmogorov consistency theorem, we classified disordered phases connected to the Ising model with mixed spin (2, 1/2) on the CT in our prior work [1]. We demonstrated that, in contrast to the single-spin Ising model [2, 3], the mixed spin Ising model has three Gibbs measures in both the ferromagnetic and anti-ferromagnetic regions. Using the self-similarity property of the infinite CT, we computed the free energy and the corresponding entropy for the single-spin Ising model [4] (see [5]). In this paper, for the first time, we will construct the Gibbs measures associated with the Ising model having mixed spin (2, 1/2) on a CT by means of the self-similarity property of the semi-infinite CT. We will establish the partition function of the model. We will compute some thermodynamic quantities of the model under null boundary condition. We will obtain some exact formulas to calculate the free energy, entropy, magnetization and susceptibility associated with the model. We will study the asymptotic behavior of these quantities especially close to the critical temperature T → 0 and when T → ∞.

István Kolossváry | A variational principle for box counting quantities

The classical variational principle for topological pressure is an essential tool in thermodynamic formalism. The aim of the talk is to extend the framework into a non-conformal setting where we prove a variational principle for an appropriate pressure that is specifically tailored to calculate box counting quantities. Similarities and differences between the variational principles will be highlighted. As an application, we can derive the L spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.

Reza Mohammadpour | Restricted variational principle of Lyapunov exponents for typical cocycles

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact. In several cases it was then possible to prove the so-called restricted variational principle formula: For every possible value α of the Lyapunov exponent, the topological entropy of the set of points with the Lyapunov exponent α is equal to the supremum of measure-theoretic entropies of invariant measures with Lyapunov exponent α.

In this talk, I will investigate the structure of the level sets of all Lyapunov exponents for typical cocycles. I will show that the restricted variational principle formula for a vector of Lyapunov exponents holds for typical cocycles. This generalizes the works of Barreira-Gelfert, and Feng-Huang.

Roope Anttila | Assouad dimension of self-affine sets and applications to Takagi functions

I will discuss recent results in the theory of Assouad dimensions of self-affine sets with emphasis on general "non-carpet-like" sets. Under a geometric separation condition, which is strictly weaker than the strong separation condition, I will provide a connection between the Assouad dimensions of self-affine sets and the Hausdorff dimensions of their slices and projections. The results will then be applied to study the slices of Takagi functions, which are a family of nowhere differentiable continuous functions. The talk is based on a joint work with Balázs Bárány and Antti Käenmäki.

Subhash Chandra | On construction of fractal functions and fractal measures

The Kannan mapping is an improvement over the contraction mapping. Barnsley (Constr. Approx. (1986) 303-329) considered the collection of self-contraction mappings and Banach fixed point theory to construct fractal functions. In this work, we give a novel method to construct fractal functions using the Kannan contraction theory with graphical illustrations. Fractal measures play an important role in the theory of fractal dimensions. We also show the existence of fractal measures supported by the attractor of the Kannan iterated function system satisfying the strong separation condition. In the last, we said some lights on fractal dimensions of the graph of fractional integrals.

Vilma OrgoványiRandom self-similar sets on the line

Motivated by the study of rational angled projection of random self-similar carpets to lines, we consider certain families of random self-similar sets on the line from the perspective of dimension, Lebesgue measure and existence of interior point. More precisely, we consider certain homogeneous families of self-similar systems on the line satisfying the so-called Finite Type Property. We define the corresponding random self-similar IFS and associate finitely many matrices to this system. In this talk I will explain the relationship between the corresponding pressure function and the above-mentioned properties of the attractor of our random self-similar system. Based on joint work with Károly Simon.

Yuki Takahashi | Invariant measures for iterated function systems with inverses

We consider Iterated Function Systems (IFS) that contain inverses in the overlapping case. We focus on the parameterized families of IFS with inverses, satisfying the transversality condition. We show that the invariant measure is absolutely continuous for a.e. parameter when the random walk entropy is greater than the Lyapunov exponent. We also show that if the random walk entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value.

Yuki Yayama | Relative pressure functions and their equilibrium states

For a subshift (X,σX) and a subadditive sequence ℱ = {log fn}n = 1 on X, we study equivalent conditions for the existence of h ∈ C(X) such that limn → ∞(1/n)∫log fndμ = ∫hdμ for every invariant measure μ on X. For a factor map π : X → Y, where (X,σX) is an irreducible shift of finite type and (Y,σY) is a subshift, applying our results and the results obtained by Cuneo on asymptotically additive sequences, we study the existence of h with regard to a subadditive sequence associated to a relative pressure function. This leads to a characterization of the existence of a certain type of continuous compensation function for a factor map between subshifts. As an application we also study factors of weak Gibbs measures.


Graccyela Salcedo | Contracting on average and uniqueness of the stationary measure for random dynamical systems

Divya Khurana | Smooth diffeomorphism with non-ergodic generic measure

Mirmukhsin Makhmudov | Multifractal Formalism from Large Deviations

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