Archive of past seminars

The study of the geometry of dynamically defined fractals has advanced significantly over the past 60 years. A central element of this theory is the concept of horseshoes.  These hyperbolic sets arise naturally in chaotic systems and exhibit remarkable fractal structures. Most studies on the geometric aspects of dynamically defined sets focus primarily on the cases  of surfaces dynamics. As expected, the situation becomes more complex in higher dimensions. 

In this talk, we will discuss some basic questions that remain unanswered in the theory of higher dimensional horseshoes. Additionally, we will explore how combinatorial tools can be utilised to extract properties of typical horseshoes without significantly losing their complexity. 

This is a joint work with Carlos G. Moreira.

Starting with the work of Hitoshi Nakada in the early 1980s, for both the metric and the arithmetic properties of various continued fraction expansion algorithms, the natural extension has played a key role in the last 40 years. In this talk, I review this, show how old results can be obtained in a short and elegant way, and also present some new results.

The new work presented is joint work with A. Bakhtawar.

It is well-known that classical statistical properties of probability preserving dynamical systems do not hold or lose their significance for systems preserving an infinite measure. In this talk I will give an overview of some recent approaches which try to solve these problems. In particular, I will discuss the pointwise asymptotic behavior of Birkhoff sums and the thermodynamic formalism for maps of the interval with indifferent fixed points. I will also present some applications to continued fraction type maps.

It is well-known that classical statistical properties of probability preserving dynamical systems do not hold or lose their significance for systems preserving an infinite measure. In this talk I will give an overview of some recent approaches which try to solve these problems. In particular, I will discuss the pointwise asymptotic behavior of Birkhoff sums and the thermodynamic formalism for maps of the interval with indifferent fixed points. I will also present some applications to continued fraction type maps.

https://www.impan.pl/~feliksp/str/250519.pdf

The classical continued fraction algorithm as well as its generalizations to higher dimensions (like, for instance, Brun's algorithm or the Jacobi-Perron algorithm) produce sequences of unimodular matrices. Our aim is to study such sequences and to discuss convergence properties of them. These convergence properties are formulated in terms of hyperbolicity properties. We want to associate Markov partitions to these continued fraction algorithms. Indeed, it has been known for a long time that we can associate a Markov partition to a single hyperbolic matrix. It turns out that it is also possible to associate Markov partitions with a ``hyperbolic’’ sequence of matrices, and, hence, with a strongly convergent continued fraction algorithm. To define such ``nonstationary’’ Markov partitions, we relate a sequence of substitutions to a sequence of matrices. This will lead to a symbolic coding of such a sequence. This is joint work with P. Arnoux, V. Berthé, M. Minervino, and W. Steiner.

I will talk about projection theorems for general families of

maps satisfying the transversality condition. We extend classical

results of this type to universal versions, holding simultaneously for

families of measures satisfying a new condition called relative

dimension separability. The setting is general enough to include

orthogonal projections and natural projections for iterated function

systems (as well as its non-autonomous and random generalizations). This allows us to provide novel applications to orthogonal projections of ergodic measures on self-conformal sets and their multifractal analysis. This is based on a joint work with Balázs Bárány and Károly Simon.

The Bergelson conjecture from 1996 asserts that the multilinear polynomial ergodic averages with commuting transformations converge pointwise almost everywhere in any measure-preserving system. This problem was recently solved affirmatively for polynomials with distinct degrees. In this talk, I will review the recent progress on this conjecture, focusing on the multilinear circle method --- a versatile new tool that combines methods from additive combinatorics and Fourier analysis, which are crucial in problems of this kind. This is based on joint work with D. Kosz, S. Peluse and J. Wright. 

Dengue fever is a typical mosquito-borne infectious disease. We propose two-strain model describing the dynamics of dengue transmission by both mosquitoes (vectors) and humans (hosts), that reads as system of 19 non-linear ODEs.  We include secondary infections (with other strain), causing a severe form of disease and take into account vector dynamics, incubation period of dengue virus and vertical transmission to larvae, i.e. possible transmission of virus from infected female mosquitoes to their offspring. Introducing hospitalized class we could capture possibility of developing DHF (dengue hemorrhagic fever) or DSS (dengue shock syndrome). Mathematical analysis of proposed model is conducted - we study existence and local stability of equilibria of the system. In the introduced model unexpected mathematical results occur, that is why we implement some modifications and analyse such simplified system. Despite the complexity of proposed models, we manage to obtain explicit results on stability of equilibria, what is not common in the case of complicated systems like ours. The presentation is based on joint work with Marcin Choiński and Urszula Skwara.

https://www.impan.pl/~feliksp/str/250414.pdf

We will study the dynamics of the exponential map in the complex plane. A well-known technique is to divide the plane into (enumerated) strips of height 2π and to code the trajectory of a point with the numbers of the strips hit by the consecutive images of that point. This divides the plane (or the Julia set) into sets of points having the same code (itinerary), and the study of those sets has been very useful in proving various results about the dynamics of the exponential map. 

However, most of the results concern only bounded itineraries. We will discuss what is known in the unbounded case and in particular present a recent result (joint with J. Horbaczewska and R. Opoka) regarding the Hausdorff dimension of a set of points with a given unbounded itinerary.

To investigate the complexity of topological dynamical systems, Köhler introduced the notion of tameness. In this talk, we will focus on the structural theorem for minimal multiple-tame systems.

A key breakthrough in the theory of tame systems came from the work of Glasner, who showed that in a minimal tame system with an invariant measure, the factor map to the maximal equicontinuous factor is almost one-to-one. Building on the characterization of tame systems via independence sets-specifically, that a system is tame if and only if it contains no essential 2-IT-tuples, as established by Kerr and Li-Fuhrmann, Glasner, Jäger, and Oertel resolved an open question of Glasner by proving that this factor map is, in fact, regular one-to-one.

We will begin by reviewing these structural results for minimal tame systems. Then we will turn to a question by Huang, Lian, Shao, and Ye: If a minimal system with an invariant measure has no essential k-IT-tuples (for some k≥3), does it admit a similar structural theorem? We will present recent progress on this question, based on joint work with Xiangtong Wang, Leiye Xu, and Shuhao Zhang.

 It is natural to consider the space of continuous functions on the real line C(R,R) as a dynamical system w.r.t. translation in time. Not surprisingly this point of view is ubiquitous in mathematics, notable examples are given by various spaces of almost periodic functions studied by Besicovitch, Bochner, Weyl,  von Neumann and others.  A celebrated result of Bebutov and Kakutani states that C(R,R) is a universal embedding space for all topological flows whose fixed point set embeds in the unit interval. In 1973 Eberlein showed that the compact  space of $1$-Lipschitz functions from  the real line to the unit interval, is a topological model for all free measurable flows. This served as a fundamental step in his proof - together with Denker - of the Jewett-Krieger theorem for flows (1974), demonstrating the usefulness of the Lipschitz representation approach. We  generalize Eberlein’s theorem to multidimensional flows, in particular giving a new proof for the one-dimensional case. This necessitates the development of the  multidimensional version of a Lipschitz representation theorem by  Gutman, Jin and Tsukamoto (2019). The theory of topological local sections  also plays a role in the proof.   Based on a joint work with Qiang Huo.

 In several dynamical systems on a real interval (doubling map with a hole, Lorenz maps, unique beta-expansions, …), the orbits are coded by binary sequences avoiding an interval. Usually, the interval of sequences is taken with respect to the lexicographic order, and we know from Labarca and Moreira (2006) when a binary shift with a lexicographic hole has positive entropy. Together with Komornik and Zou (2024), I gave a “self-similar” description of this result in terms of substitutions (or renormalisations). The aim of this talk is to extend the characterisation of positive entropy to other piecewise monotonic orders on sequences, namely the alternating lexicographic order, the tent map order and its inverse (which correspond to the negative doubling map, the tent map and the V map). We obtain again a characterisation in terms of compositions of a finite set of substitutions, but now we have a graph-directed construction instead of arbitrary compositions of substitutions. Moreover, renormalisation for one piecewise monotonic order can lead to another one, so that the characterisations for the 4 orders are interconnected.

 PDF file: https://www.impan.pl/~feliksp/str/250310.pdf

Let $G \curvearrowright X$ be a free action of an amenable group on a compact metrizable space. The shift embeddability problem asks whether there exists a $G$-equivariant embedding of $X$ into the $M$-cubical shift $([0, 1]^M)^G$, and over the years it has become one of the central questions 

in the mean dimension theory. The best-known result to date was the theorem of Gutman, Qiao, and Tsukamoto, which

provides a satisfying answer for actions of \mathbb{Z}^d. In this talk, we will describe how this theorem can be generalized to actions of groups from a larger class. After stating the main results, we will focus on the parts of the proof that required a different approach from that of GQT, specifically:

1) defining the right notion of a tiling,

2) encoding the tilings using a small amount of information, and

3 )obtaining the tiling from an a priori weaker property.

 PDF file: https://www.impan.pl/~feliksp/str/250303.pdf

A topological dynamical system is a pair (X, T) consisting of a compact metric space X and a continuous self-map T of X. The Besicovitch pseudometric provides a different way of measuring the ”closeness” of two orbits of a topological dynamical system that focuses on long-term behavior. This pseudometric is particularly useful for studying the statistical properties of orbits, especially invariant measures generated by these orbits. For symbolic systems (subshifts) the Besicovitch pseudometric is uniformly equivalent to the pseudometric ¯d that measures the asymptotic density of differences between two sequences of symbols. During my talk, I will present the results on the Besicovitch and ¯d pseudometrics obtained in the course of my PhD studies and contained in the article [2], and in preprints [1, 3]. We study limits of sequences of generic points with respect to the Besicovitch pseudometric D_B. Using the characterization of the spectrum of an ergodic measure via its generic points in [1], we analyze the properties of the (limit) measure by passing to the limit in D_B. Specifically, we show that the set of generic points for discrete spectrum, totally ergodic, (weakly) mixing, and zero entropy measures forms a closed set with respect to the Besicovitch pseudo-metric. We also discuss joint results on the pseudo-metric ¯d on shift spaces that are presented in [2]. We also study connections between the asymptotic average shadowing property (a variant of the classical shadowing property) and the vague specification property (a variant of the classical specification property) for general topological dynamical systems. Using the Besicovitch pseudo-metric and its relatives we show that the asymptotic average shadowing property and the vague specification property are equivalent.

References

[1] S. Babel, M. Can, D. Kwietniak, P. Oprocha, Spectrum of invariant measures via generic points, preprint.

[2] M Can, J. Konieczny, M. Kupsa, and D. Kwietniak, Minimal and proximal examples of ¯d-stable and ¯d-approachable shift spaces, Ergodic Theory and Dynamical Systems, 2025;45(2):396-426. doi:10.1017/etds.2024.43.

[3] M. Can, A. Trilles, The equivalence of asymptotic average shadowing and vague specification properties and its consequences, [arXiv:2411.01556], preprint.

Smale spaces were defined by David Ruelle in the 1970's as topological models for the typically fractal-like hyperbolic nonwandering sets of Stephen Smale's Axiom A systems. A Smale space is a compact metric space together with a homeomorphism implementing hyperbolic dynamics and a local product structure. Prototype examples are the topological Markov chains, aperiodic substitution tilings and hyperbolic toral automorphisms.  This talk will give an example-driven introduction to Smale spaces with a focus on their dimension theory, which can be studied using Markov partitions and Ahlfors regular measures. 

Finally, I will briefly mention how the dimension theory of a Smale space is related to fine analytic properties of the operator algebras encoding the stable and unstable foliations on it.

There are many different ways that one can define fractal dimension. The three of interest in this talk will be Hausdorff dimension, lower box dimension, and upper box dimension. It is well known that these notions of dimension can all be different for general sets, but they coincide for many classes of fractal sets such as self-similar sets. This raises the question of what conditions one needs to assume about an iterated function system (IFS) to be sure that all dimensions will coincide for the resulting fractal? We will survey some results and open problems related to this broad question in three specific settings, namely affine IFSs, bi-Lipschitz IFSs, and infinite conformal IFSs. This talk is based on a joint project with Simon Baker, De-Jun Feng, Chun-Kit Lai and Ying Xiong, and another paper with Alex Rutar. 

In dimension three, we present three conservative partially hyperbolic settings (derived from Anosov systems, time one-maps of geodesic flows on negatively curved surfaces, and skew-product with circle fibers) where the following dichotomy holds: the diffeomorphism is either Anosov or it supports and (ergodic) nonhyperbolic measure. 

We also see that the second possibility is quite frequent.

Joint work with J, Yang (UFF, Niterói, Brazil) J. Zhang (Beihaan, Beijing, China).

New dynamical characterizations for the (T)- and Haagerup property  for locally compact Polish groups are found via nonsingular Poisson actions of various Krieger's types.

For partially hyperbolic diffeomorphisms with minimal strong foliations and unstable/stable blender-horseshoes, we establish restricted variational principles for entropy, fixing a specified center Lyapunov exponent and varying the metric entropies among ergodic measures. We prove that for each exponent value in the interior of the spectrum (including value 0), every possible entropy value can be achieved by some ergodic measure. This is joint work with LD Díaz, M Rams, and J Zhang.

 In this talk, we present a surgery construction that replaces the interior dynamics in an orbit of wandering domains with the non-autonomous dynamics of a sequence of Blaschke products, as long as both are uniformly hyperbolic. The surgery is performed in infinitely many domains at once, despite mapping to each other with a degree larger than one. As an application, we construct an entire function with a wandering domain for which discrete and indiscrete grand orbit relations coexist, in a way that is not possible for periodic Fatou components. Understanding grand orbit relations in the different types of Fatou components is a key step in the study of quasiconformal deformations of holomorphic maps.

This is joint work with Vasiliki Evdoridou, Lukas Geyer and Leticia Pardo-Simon.

The main goal of my talk is to analyze some peculiar features of the global (and local) minima of α-Brjuno functions Bα where α ∈ (0, 1]. Our starting point is the result by Balazard–Martin (2020), who showed that the minimum of B1 is attained at the golden number g.

We shall refine this result in two directions: we consider the problem of characterizing local minima of B1 and  we consider the problem of characterizing global and local minima of Bα for other values of α.

Kac’s lemma is a classical result in ergodic theory. It asserts that the expected number of iterates that it takes a point from a measurable set A to return to the set A under an ergodic probability-preserving transformation is equal to the inverse of the measure of A.   As we will discuss in this seminar, there is a natural generalization of Kac’s lemma that applies to probability preserving actions of an arbitrary countable group (and beyond). As an application, we will show that any ergodic action of a countable group admits a countable generator. The content of this work is based on a joint article with Benjamin Weiss.

Last time I was speaking about iterated function systems with randomized translations, this time we will randomize contraction ratios. 

This leads to interesting new phenomena, in particular this time the Hausdorff dimension is not constant even in the OSC case, and it turns out that its almost sure value depends on the way the randomness is introduced. 

I plan to present the actual state of the area plus new results we have 

with Balazs Barany and Antti Kaenmaki.

Consider a random linear iterated function system on the line, with fixed contractions and random translations. 

This type of systems was studied, with Jordan, Pollicott, and Simon proving that when the similarity dimension $s<1$ then the Hausdorff dimension of the attractor is almost surely equal to $s$, and if $s>1$ then the attractor 

has almost surely positive Lebesgue measure. The latter case was recently improved by Dekking, Simon, Szekely, and Szekeres, who proved that if $s>1$ then the attractor almost surely contains an open interval.

In this talk I will present our latest work with Balazs Barany, in which we prove that actually, if $s>1$ and under some non-restrictive conditions on the probability distribution, the density of the (random) natural measure is Holder continuous. The proof uses an approach we learned from a paper of Erraoui and Hakiki on fractional Brownian motions.

We study the variations of the dimensions (Hausdorff and packing) of harmonic measure of non-autonomous conformal IFS and we establish continuity results.

In 2020 Boyland, de Carvalho and Hall published a seminal paper [6] in which they provided a detailed analysis of boundary dynamics of a parameterized family of sphere homeomorphisms with attractors homeomorphic to the (core) tent inverse limit spaces through the study of prime ends. This result presented the first such study of a chaotic parameterized family on a surface. Prior to this result much attention in Continuum Theory has been directed towards the topological classication of tent inverse limit spaces with the highlight being the result [1] that for any two different parameters in the parameter range (\sqrt2; 2] the tent inverse limits (not restricted to its dynamical core) are non-homeomorphic. Motivated by these result we constructed two parameterized families of sphere omeomorphisms varying continuously with the parameter in [0; 1/2] with attractors all homeomorphic to the pseudo-arc and the pseudo-circle respectively, yet presenting rich boundary dynamics [4, 5]. In this talk I will present these constructions that rely on a very useful technique called BBM (Brown-Barge-Martin), which incorporates inverse limits and natural extensions of the underlying bonding maps to embed attractors in manifolds and was presented by Boyland, de Carvalho and Hall in [7]. I will address how this study is intertwined with the study of typical properties of one-dimensional Lebesgue measure-preserving maps on one-dimensional manifolds [2, 3]. This talk is based on joint works with Piotr Oprocha (AGH Krakow & University of Ostrava) as well as Jozef Bobok (CVUT Prague) and Serge Troubetzkoy (Aix Marseille).

References

[1] M. Barge, H. Bruin, S. Stimac, The Ingram Conjecture, Geom. Topol. 16 (2012), 2481-2516.

[2] J. Bobok, J. Cinc, P. Oprocha, S. Troubetzkoy, Are generic dynamical properties stable under composition with rotations? Proc. Amer. Math. Soc., 152 (2024), 3011-3026.

[3] J. Bobok, J. Cinc, P. Oprocha, S. Troubetzkoy, The generic Lebesgue measure-preserving map is invertible a.e., arXiv:2405.09917, May 2024.

[4] J. Cinc, P. Oprocha, Parameterized family of annular homeomorphisms with pseudo-circle attractors, Journal of Differential Equations 407 (2024), 102-132.

[5] J. Cinc, P. Oprocha, Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations, Proc. Lond. Math. Soc. 125 (2022), 318-357.

[6] P. Boyland, A. de Carvalho, T. Hall, Inverse limits as attractors in parametrized families, Bull. Lond. Math. Soc. 45, no. 5 (2013), 1075-1085.

[7] P. Boyland, A. de Carvalho, T. Hall, Natural extensions of unimodal maps: prime ends of planar embeddings and semi-conjugacy to sphere homeomorphisms, Geom. Topol. 25 (2021) 111-228.

Consider a countable discrete group  Γ and its subgroup space-Sub(Γ), the collection of all subgroups of Γ.  Sub(Γ) is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on {0, 1}^Γ). The normal subgroups of Γ are the fixed points of (Sub(Γ), Γ). Furthermore, the Γ-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs).

Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. Among many other remarkable results, they strengthen the well-known Margulis’s normal subgroup Theorem.

More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of sub algebras of L(Γ).

Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of L(Γ). In this talk, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.

 We will present a four terms asymptotic formula, we have obtained, of the Hausdorff dimension of the radial Julia set $J_r(f_{\lambda})$ for the exponential function $f_{\lambda}(z)=\lambda e^z$ as $\lambda \to 0$.  Moreover, we will also discuss the asymptotic behaviour of the Hausdorff dimension of the radial Julia set of cosine functions $f_{a,b}(z)=ae^z+be^{-z}$ as $a,b \to 0$ in various ways. This is a joint work with Prof. Qiu.

 After recalling some basic notions in Analytic Group Theory, I will discuss a possible approach to the problem of determining whether two properties which represent a weak form of hyperbolicity, namely the (AO)-property and bi-exactness, coincide. I will also explain how the techniques involved can be used in order to study certain analytic properties of SL(3,Z) (joint with F. Radulescu). I will focus on the role of measurable dynamics and proximality arguments in this context.

 Kolmogorov's covers were introduced in Kolmogorov's celebrated  theorem on superpositions  of continuous functions.  The main goal of the talk is to show how Kolmogorov's covers can be used  for constructing finite-to-one equivariant maps to cubical shifts.

 The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this operator is intrinsically defined, its spectral properties are analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden). If time allows, I will also discuss the recent joint work with Magnus Goffeng (Lund) and Bram Mesland on applying the log-Laplacian to study the spectral geometry of Cuntz-Krieger algebras. The latter are C*-algebras associated with stable/unstable foliations on topological Markov chains.

Ergodic optimization is the study of problems relating to maximizing invariant measures and maximum ergodic averages. In ergodic optimization theory, one important problem is the typically periodic optimization (TPO) conjecture. This conjecture was proposed by Mañé [6], Hunt, Ott and Yuan in the 1990s, which reveals the principle of least action in dynamical system settings. To be more precise, TPO indicates that when the dynamical system is suitably hyperbolic and the observable is suitably regular, then the maximizing measure is "genetically" supported on a periodic orbit with relatively low period. In the setting of uniformly open expanding maps with Lipschitz/Holder observables, TPO was obtained in topological genetic sense by Contreras in 2016. In this talk, I will report a numer recent progress on understanding TPO conjecture both in probabilistic and topological sense, and for more general uniformly and non-uniformly hyperbolic systems.

An analytic endomorphism of the unit disk is called an inner function if its boundary limit defines a transformation of the circle - which is necessarily Lebesgue nonsingular. I'll review the ergodic theory of inner functions & then present results on their:

• structure (preimage multiplicities vs singularity set);

• transfer operators (spectral gaps and perturbations);

• a conditional central limit theorem for their stochastic processes.

If there's still time at the end I'll mention some questions concerning the multidimensional situation. Based on joint work with Mahendra Nadkarni: arXiv:2305.15278.

The talk is joint with the mini-conference Ergodic group actions and unitary representations at IMPAN-BC. 

The notion of Borel reducibility, developed mainly by descriptive set theorists, is a tool for measuring the complexity of an equivalence relation. Given a class of mathematical objects, it can usually be applied to study the isomorphism relation between those objects. In 2011, Foreman, Rudolph and Weiss showed that the conjugacy of of ergodic MPS (measure-preserving systems) is not Borel. The result can be interpreted as proof of non-existence of certain simpler descriptions of when two ergodic MPS are isomorphic (for example, no complete invariant exists). Several other classes of topological and measurable dynamical systems have also been studied in a similar manner. We will provide an introduction to the topic of Borel reducibility, survey the results related to dynamical systems, and discuss some new results. 

Full groups originated from the theory of measurable (and later Cantor) dynamical systems and their von Neumann-algebra (C*-algebra) crossed-products. For a given topological dynamical system (X,G), the full group [G] can be broadly defined as the set of all homeomorphisms of X that act within the G-orbits. Thus, the full groups can be viewed as a generalized symmetric group of the orbit equivalence relation of (X,G). In a series of papers by Giordano-Putnam-Skau, Matui, Medynets, Nekrashevych, and others, it was shown that full groups (as abstract groups) encode complete information about the underlying dynamical systems up to (topological) orbit equivalence. In recent years, the development of the theory of full groups for Cantor minimal systems has been having considerable impact on geometric group theory driven primarily by the fact that by tweaking dynamical properties of the underlying dynamical system (X,G), we can produce a (countable) full group [G] with new and unusual properties, which has been successfully used to solve some open problems in geometric group theory.

The goal of our talk is to give a gentle introduction into the theory of full groups, discuss known results, and present open problems. 

We consider the environment viewed by the particle process in a random walk in a quasiperiodic environment. This process appears to be a simply defined random walk on the circle. We prove the central limit theorem and establish the rate of mixing in the Diophantine case. This is based on a joint work with D. Dolgopyat.

Maximal functions, especially of the Hardy-Littlewood type, are one of the most important objects of study in harmonic analysis. Their Lp boundedness has been known for many decades. However, the exact values of their Lp norms are mostly unknown and already obtaining an estimate for these norms is a challenging question. In the talk I will discuss discrete counterparts of these maximal functions defined over the d-dimensional integer lattice. Systematic study of dimension-free estimates for such operators has been initiated several years ago in collaboration with Bourgain, Mirek, and Stein. I will overview existing results and present recent progress in the field. I will also mention connections to ergodic theory established via the Calderón transference principle.

The talk will begin by providing a brief review of entropy, preimage entropy, and metric mean dimension from both the measure-theoretic and topological perspectives. Then, we will introduce the preimage metric mean dimension, which is introduced for studying systems with infinite preimage entropy. Specifically, we will prove a partial variational principle and compute the preimage mean metric dimension for a series of examples. Based on joint work with Fagner B. Rodrigues.

The local connectivity of the boundary of a simply connected Fatou component U allows us to understand the dynamics on the closure of U. For transcendental entire maps an unbounded non-univalent Fatou component can never have a locally connected boundary. In this talk we prove local connectivity of boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. These basin boundaries, which may be unbounded, can contain singular values as well as the essential singularity at infinity but we assume that their unbounded parts are contained in regions where the map exhibits a kind of 'parabolic' behavior.

The talk is based on a joint work with Krzysztof Barański, Nuria Fagella and Xavier Jarque. 

I will speak about the new Boris Solomyak's article, in which he presents an example of a conformal iterated function system on the line in which two of the maps share a common fixpoint, but still there is no dimension drop.

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 recent joint work with Carlos Gustavo Moreira and Carlos Matheus Santos (arxiv:2212.11371). 

In 1970s Bowen related hyperbolic dynamics with specification property and used this to show that there exists a unique measure of maximal entropy. Almost the same time Sigmund used specification property as a tool in characterization of simplex of invariant measures. Since then, these results were inspiration for numerous mathematicians in various studies of dynamics. Several weaker versions of specification property were developed and used as a tool for better understanding of dynamics. At the same time, questions, how often such properties can be found in dynamics were raised (e.g. in the sense of Baire category theorem).

In this talk we will present selected questions and results fitting into the above framework of research.

According to the classical Menger–Nöbeling (1932) theorem, a compact metric space X of (Lebesgue covering) dimension less than r/2 admits a topological embedding into r-dimensional Euclidean space. Generalizing this to the dynamical setting we prove that whenever an (arbitrary) group G acts on a finite-dimensional compact metric space X, there exists an equivariant topological embedding of X into ([0,1]r)G, provided that for every positive integer N, the dimension of the space of points in X with orbit size at most N is strictly less than Nr/2. Note that the equivariant topological embedding is necessarily of the form x↦(f(gx))g∈G for some continuous map f:X→[0,1]r. Going further we derive a topological Takens theorem for finitely generated group action, that is under the assumptions above when the group G is finitely generated one may find a continuous map f:X→[0,1]r so that x↦(f(gx))g∈G′ is injective for some finite G′⊂G where the cardinality of G′ is bounded by a function of r, the dimension of X and the number of generators of G.

Based on a joint work with Michael Levin and Tom Meyerovitch.

It all started in 1889 with F. Engel's article on the (2,3,4)-distributions. The language was PDE's. Engel proved that the couple of differential equations in 4 variables having generic properties (in contemporary language, a rank-2 distribution span(X,Y) in the tangent bundle to a 4-manifold such that dimspan(X,Y,[X,Y])=3 everywhere and dimspan(X,Y,[X,Y],[X,[X,Y]],[Y,[X,Y]])=4 everywhere) behaves LOCALLY in a unique way.

E. von Weber continued in 1898 putting forward a condition now called the Goursat condition: IF the derived growth (2,3,4,…,n−1,n) occurs locally everywhere, THEN a unique model couple of PDE's in n variables occurs. This Weber's theorem was grossly false, because it implied that the Goursat condition was locally trivial and had no singularities whatsoever. Yet von Weber was a pioneer.

E. Cartan (1914) started to geometrize, Goursat (1922) widely popularized. A true depth of the Goursat condition was discovered only in 1978 (Giaro-Kumpera-Ruiz). These authors produced a singular behaviour within the derived growth (2,3,4,5). A renewal of interest in the singularities of Goursat structures has been taking place since 1996 until now. Since 1999 (Kumpera-Rubin) other regular derived growths have attracted interest, in particular (3,5,7,…,2m−1,2m+1) everywhere, i.e., so-called SPECIAL 2-MONSTERS.

Attempts at local classification of them will be outlined, terminating in mentioning an ISSUE OF INTEREST to the dynamical systems' community: local diffeomorphisms of R3

that are not embeddable in flows of smooth autonomous vector fields. The latter are badly obstructing the classifications. A 2010' result says that the derived growth (3,5,7,9,11) has exactly 34 local geometric realizations. A challenging difficult issue is the derived growth (3,5,7,9,11,13). The speaker, helped recently by A. Weber, hopes for a discrete/finite? local classification of those special 2-monsters of length 5.

Abstract

Let G be an infinitely countable amenable group and let K be a finite subset of G containing the unit and at least one more element. A subset C of G is K-separated if the sets Kc are disjoint as c ranges over C. A K-separated set C is maximal if no proper superset of C is K-separated. The collection of the indicator functions of all maximal K-separated sets is closed and shift-invariant. We call it the K-shift. Last year, Benjy Weiss proposed to show that K-shifts are universal in the sense that any free ergodic measure-preserving action whose entropy is less than that of the K-shift is isomorphic to some invariant measure on the K-shift. Together with Benjy, Mateusz Więcek and Guohua Zhang we are trying to prove this. However an unexpected obstacle has been revealed (exactly two days ago) which makes the subject even more interesting.

Diophantine Approximation is a branch of Number theory in which the central theme is understanding how well real numbers can be approximated by rationals. Dirichlet's theorem (1842) is a fundamental result that gives an optimal approximation rate of any irrational number. The set of real numbers for which Dirichlet's theorem admits an improvement was originally studied by Davenport and Schmidt. It has been recently proved that the improvements to Dirichlet's theorem are related to the growth of the products of consecutive partial quotients. In this talk I will discuss some new metrical results for the set of Dirichlet non-improvable numbers in connection with the theory of continued fractions.

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