International conference held online at UniwersytetL´odzki, L´od´z,Poland

BOOK OF ABSTRACTS

1 Dynamics of (Semi–)Group Actions

The modern theory of dynamical systems encompasses the study of group and semi-group actions, which include vibrant, exciting, and very active lines of research. The primary aim of the conference is to bring together mathematicians involved in topological, mea- surable and algebraic aspects of dynamical systems generated by (semi-)group actions, to stimulate the exchange of new ideas and to offer new opportunities for broadening scientific contacts. This is reflected by the inclusion of young and experienced researchers giving plenary talks and contributed talks. The scope of this conference encompasses research of the dynamical, geometric and algebraic aspects of semi-group actions by continuous mappings of real or complex variables. The wide range of topics include thermodynamic formalism for random dynamical systems, fractal dimension of self-similar sets, entropy and pressure functionals for semigroup actions, topological and ergodic aspects of algebraic actions, among many others. This is the second edition of the conference (the first one took place inL´od´zon 2019). We wish you a pleasant and fruitful (even if online) participation.

Organizers

2 Organizers

Scientific Committee

Andrzej Bi´s(UniwersytetL´odzki,Poland)

Dikran Dikranjan (Universit`adegli Studi di Udine, Italy)

Tomasz Downarowicz (Politechnika Wroc lawska, Poland)

Lubomir Snoha (Univerzita Mateja Bela, Slovakia)

Luchezar Stoyanov (University of Western Australia, Australia)

Mariusz Urba´nski(University of North Texas, USA)

Paulo Varandas (Universidade Federal da Bahia, Brazil; Universidade do Porto, Portugal)

Organizing Committee

Andrzej Bi´s(UniwersytetL´odzki)— chairman

Kamil Niedzia lomski(UniwersytetL´odzki)— secretary

Marek Badura (UniwersytetL´odzki)

Adam Bartoszek (UniwersytetL´odzki)

Magorzata Ciska–Niedzialomska (UniwersytetL´odzki)

Maciej Czarnecki (UniwersytetL´odzki)

Yonatan Gutman (Instytut Matematyczny Polskiej Akademii Nauk)

Marzena Jaworska-Banert (UniwersytetL´odzki)

Wojciech Kozlowski (UniwersytetL´odzki)

Ryszard Pawlak (UniwersytetL´odzki)

Sponsors

UniwersytetL´odzki

WydzialMatematyki i Informatyki UniwersytetuL´odzkiego

3 Schedule

Hours of talks in the European summer time (UTC+2)

Tuesday, June 22

09:50 – 10:00 Opening

10:00 – 10:50 Mark Pollicott, Rate of escape for random walks on SL(2, R) 11:00 – 11:50 MichalRams, Restricted variational principle for partially hyperbolic dynamics 12:00 – 12:30 coffee break 12:30 – 12:55 Antongiulio Fornasiero, Algebraic entropy and Hilbert polynomials 13:00 – 13:25 Jose C´anovas, On two notions of fuzzy topological entropy 13:30 – 13:55 Andrei Tetenov, Three theorems on self-similar continua with finite intersection property 14:00 – 16:00 lunch break 16:00 – 16:50 Maria Carvalho, Generalized entropy maps, variational principles and equilibrium states 17:00 – 17:50 Eugen Mihailescu, Pressure functionals in semigroup dynamics 18:00 – 18:25 Ryszard Pawlak, On points of distributive chaos 18:30 – 18:55 Jason Atnip, Thermodynamic formalism for random interval maps with holes

Wednesday, June 23

10:00 – 10:50 Eli Glasner, Topological characteristic factors and nilsystems 11:00 – 11:50 Alexander Bufetov, Determinantal point processes: quasi-symmetries and interpolation 12:00 – 12:30 coffee break 12:30 – 12:55 Nikolai Edeko, A dynamical proof of the van der Corput inequality 13:00 – 13:25 Lubomir Snoha, Product of minimal spaces: complete solution 13:30 – 13:55 Everaldo de Mello Bonotto, Global attractors for semigroups under impulse action 14:00 – 16:00 lunch break 16:00 – 16:50 Vitaly Bergelson, Dynamics actions of (N, ×) 17:00 – 17:50 Vadim Kaimanovich, Coincidence, equivalence and singularity of harmonic measures 18:00 – 18:25 Henrik Kreidler, The Furstenberg-Zimmer structure theorem revisited 18:30 – 18:55 Konstantin Slutsky, L1 full groups of measure-preservingactions of Polish normed groups

4 Thursday, June 24

10:00 – 10:25 Takatyuki Watanabe, Non-i.i.d. random holomorphic dynamical systems 10:30 – 10:55 Johannes Jaerisch, Hausdorff dimension of escaping sets for Z-extensions of expanding interval maps 11:00 – 11:25 Andrzej Bi´s, Topological entropy, upper Carath´eodory capacity and fractal dimensions of semi- group actions 11:30 – 11:55 Till Hauser, Entropy beyond actions of uniform lattices 12:00 – 12:30 coffee break 12:30 – 12:55 Yuto Nakajima, The Hausdorff dimension of some planar sets with unbounded digits 13:00 – 13:50 Manuel Stadlbauer, Equilibrium states associated with the joint action of Ruelle expanding maps 14:00 – 16:00 lunch break 16:00 – 16:50 Anna Giordano Bruno, Receptive topological and metric entropy 17:00 – 17:50 Tom Ward, Order of mixing in algebraic systems 18:00 – 18:25 Mariusz Urba´nski, Random non-hyperbolic exponential maps 18:30 – 18:55 Mauricio D´ıaz, Levels of local chaos for special Blocks Families

Friday, June 25

10:00 – 10:50 Hiroki Sumi, Random dynamical systems of regular polynomial maps on C2 11:00 – 11:50 Benjamin Weiss, The Foias-Stratila theorem in higher dimensions 12:00 – 12:30 coffee break 12:30 – 12:55 Oleg Gutik, On some generalization of the bicyclic monoid 13:00 – 13:25 Adam Spiewak,´ Singular stationary measures for random piecewise affine interval homeomor- phisms 13:30 – 15:30 lunch break 15:30 – 16:20 Raimundo Brice˜no, Kieffer–Pinsker type formulas for Gibbs measures 16:30 – 17:20 Feliks Przytycki, On Hausdorff dimension of polynomial not totally disconnected Julia sets 17:30 – 17:45 Closing

5 Abstracts

Tuesday, June 22, 18:30-18:55 Jason Atnip University of South Wales, Australia [email protected] Thermodynamic formalism for random interval maps with holes

In this talk we will consider a collection of piecewise monotone interval maps, which we iterate randomly, together with a collection of holes placed randomly throughout phase space. Birkhoffs Ergodic Theorem implies that the trajec- tory of almost every point will eventually land in one of these holes. We prove the existence of an absolutely continuous conditionally invariant measure, conditioned according to survival from the infinite past. Absolute continuity is with respect to a conformal measure on the closed systems without holes. Furthermore, we prove that the rate at which mass escapes from phase space is equal to the difference in the expected pressures of the closed and open systems. Finally, we prove a formula for the Hausdorff dimension of the fractal set of points whose trajectories never land in a hole in terms of the expected pressure function.

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Wednesday, June 23, 16:00-16:50 Vitaly Bergelson Ohio State University, USA [email protected] Dynamics of actions of (N, ×)

We will present several examples of actions of the multiplicative semigroup (N, ×) and discuss some interesting applications to Ramsey theory and Diophantine approximations. We will also discuss and juxtapose combinatorial and Diophantine properties of normal sets in semigroups (N, +) and (N, ×). We will conclude with a brief review of some interesting open problems.

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Thursday, June 24, 11:00-11:25 Andrzej Bi´s UniwersytetL´odzki,Poland [email protected] Topologigal entropy, upper Carath´eodory capacity and fractal dimensions of semigroup actions

We study dynamical systems given by the action T : G × X → X of a finitely generated semigroup G with identity 1 on a compact metric space X by continuous selfmaps . For any finite generating set G1 of G, the receptive topological entropy of G1 (in the sense of Ghys et al. (1988) and Hofmann and Stoyanov (1995)) is shown to coincide with the limit of upper capacities of dynamically defined Carath´eodory structures on X depending on G1, and a similar result holds

6 true for the classical topological entropy when G is amenable. Moreover, the receptive topological entropy and the topological entropy of G1 are lower bounded by respective generalizations of Katoks δ–measure entropy, for δ ∈ (0, 1). In the case when T (g, −) is a locally expanding selfmap of X for every g ∈ G \{1}, we show that the receptive topological entropy of G1 dominates the Hausdorff dimension of X modulo a factor log λ determined by the expanding coefficients of the elements of {T (g, −): g ∈ G1 \{1}}. The talk is based on joint work with Dikran Dikranjan (Udine), Anna Giordano Bruno (Udine) and Luchezar Stoyanov (Perth).

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Wednesday, June 23, 13:30-13:55 Everaldo de Mello Bonotto Universidade de S˜aoPaulo, Brazil [email protected] Global attractors for semigroups under impulse action

Impulsive semidynamical systems are used to describe the evolution of processes whose continuous dynamics are interrupted by abrupt changes of state. In this talk, we present the theory of impulsive semidynamical systems and we exhibit the definition of global attractors for this class of impulsive systems. Sufficient conditions are presented to guarantee the existence of a global attractor. [1] E. M. Bonotto; M. C. Bortolan; A. N. Carvalho; R. Czaja, Global attractors for impulsive dynamical systems – a precompact approach, Journal of Differential Equations, 2602-2625, 2015.

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Friday, June 25, 15:30–16:20 Raimundo Brice˜no Poinficia Universidad Cat´olicade Chile, Chile [email protected] Kieffer–Pinsker type formulas for Gibbs measures

In this talk we will discuss some new results regarding expressions for entropy and pressure in the context of Gibbs measures defined over countable groups. Our starting point will be the Pinsker formula for the Kolmogorov- Sinai entropy of measure preserving actions of orderable amenable groups. Then, we will consider a formula for pressure that was developed by Marcus and Pavlov (2015). Next, we will review some techniques based on random orderings, mixing properties of Markov random fields, and percolation theory in order to generalize previous work. Time permitting, we will discuss some applications and establish connections between these results and part of the role that phase transitions play in our understanding of entropy.

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Wednesday, June 23, 11:00-11:50 Alexander Bufetov Universite´ed’Aix–Marseille & CNRS, France [email protected] Determinantal point processes: quasi–symetries and interpolation

The study of point processes, that is, random subsets of a Polish space, goes back at least to the 1662 work of John Graunt on mortality in London. Matrices whose entries are given by chance were studied by Ronald Fisher in 1915 and John Wishart in 1928 and used by Freeman Dyson who in 1962 observed that the statistical theory (...) will describe the degree of irregularity (...) expected to occur in any nucleus.

7 The Weyl character formula implies that the correlation functions for the eigenvalues of a Haar-random unitary matrix have determinantal form — the Ginibre-Mehta theorem, — and in 1973 Odile Macchi started the systematic study of point processes whose correlation functions are given by determinants. This level of abstraction has proved very fruitful: on the one hand, examples of determinantal point processes arise in diverse areas such as asymptotic combinatorics (Burton-Pemantle, Benjamini-Lyons-Peres-Schramm, Baik-Deift- Johansson, Borodin-Okounkov-Olshanski), representation theory of infinite-dimensional groups (Olshanski, Borodin- Olshanski), random series (Hough- Krishnapur-Peres-Vir´ag)and, of course, random matrices; on the other hand, the general theory of determinantal point processes includes limit theorems (Soshnikov), a characterization of Palm measures (Shirai-Takahashi), the Kolmogorov as well as the Bernoulli property (Lyons, Lyons-Steif), and rigidity (Ghosh, Ghosh-Peres). In this talk, the correlation kernels of our determinantal point processes will be assumed to induce orthogonal projections: for example, the sine-kernel of Dyson induces the projection onto the Paley-Wiener space of functions whose Fourier transform is supported on the unit interval, while the Bessel kernel of Tracy and Widom induces the orthogonal projection onto the subspace of square-integrable functions whose Hankel transform is supported on the unit interval. What is the relation between the point process and the Hilbert space that governs it? Extending earlier work of Lyons and Ghosh, in joint work with Qiu and Shamov it is proved that almost every realization of a determinantal point process is a uniqueness set for the underlying Hilbert space. For the sine-process, almost every realization with one particle removed is a complete and minimal set for the Paley-Wiener space, whereas if two particles are removed, then one obtains a zero set for the Paley-Wiener space. Quasi-invariance of the sine-process under compactly supported diffeomorphisms of the line plays a key rle. The 1933 Kotelnikov theorem samples a Paley-Wiener function from its restriction onto the integers. How to reconstruct a Paley-Wiener function from a realization of the sine-process? In joint work with Borichev and Klimenko, it is proved that if a Paley-Wiener function decays at infinity as a sufficiently high negative power of the distance to the origin, then the Lagrange interpolation formula yields the desired reconstruction. Similar results are also obtained for the Airy kernel, the Bessel kernel and the Ginibre kernel of orthogonal projection onto the Fock space. In joint work with Qiu, the Patterson-Sullivan construction is used to interpolate Bergman functions from a realization of the determinantal point process with the Bessel kernel, in other words, by the Peres-Virg theorem, the zero set of a random series with independent complex Gaussian entries. The invariance of the zero set under the isometries of the Lobachevsky plane plays a key role.

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Tuesday, June 22, 13:0-13:25 Jose C´anovas Universidad Polit´ecnicade Cartagena, Spain [email protected] On two notions of fuzzy topological entropy

We explore the notion of fuzzy topological entropy when different definitions of fuzzy compactness are considered. We prove that the definitions by Tok [2] and by Uzzal Afsan and Basu [3] are useless since they always display zero entropy. We give a simple proof of the bridge result which state that topological entropy agrees with fuzzy topological entropy when it is defined by using Lowen’s definition of fuzzy compactness [1], [4]. The particular case of interval maps is also discussed, proving that the fuzzy entropy also agrees with the topological entropy of the Zadeh fuzzification of the crisp map on the set of convex and normal fuzzy sets. [1] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 18 (1976), 145-174. [2] I. Tok, On the fuzzy topological entropy function, JFS 28 (2005), 74-80. [3] R.M. Uzzal Afsan, C.K. Basu, Fuzzy topological entropy of fuzzy continuous functions on fuzzy topological spaces, Applied Mathematical Letters 24 (2011), 2030-2033. [4] B M Uzzal Afsan, Weakly fuzzy topological entropy, preprint (2020).

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8 Tuesday, June 22, 16:00-16:50 Maria Carvalho Universidade do Porto, Portugal [email protected] Generalized entropy maps, variational principles and equilibrium states

In this talk, based on joint work with Andrzej Bi´s(University of Lodz), Miguel Mendes (University of Porto) and Paulo Varandas (University of Bahia and CMUP), I will present a unifying approach to the thermodynamic formalism of dynamical systems and semigroup actions. More precisely, for each pressure function we construct an upper semi- continuous affine entropy-like map which, in the context of continuous transformations of a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy. This strategy provides an abstract variational principle and guarantees that equilibrium states always exist, though they may be at best finitely additive. Afterwards, I will discuss some applications.

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Thursday, June 24, 18:30-18:55 Mauricio D´ıaz Universidad del B´ıoB´ıo,Chile [email protected] Levels of local chaos for special Blocks Families

In this work we going to study the properties of some chaotic properties via Furstenberg families, specially using other levels of Blocks families. Futhermore, We are going to relate the Li Yorke and Distribution chaos levels through the existence of certain categories of block families, which contain IP families and Weakly Thick families. Then, we are going to demonstrate that there is equivalence between chaotic localities in time actions that are closed under addition and multiplication, showing at the end some applications in types of shift systems from an ergodic point of view. [1] Li, J., Transitive points via Furstenberg family, Topology and its Applications 158 (16) (2011) 2221–2231. [2] Shao, S., Proximity and distality via Furstenberg families, Topology and its Applications, 153(12) (2006) 2055–2072. [3] Pawlak, R.J., Loranty, A., On the local aspects of distributional chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(1) (2019) 013–104. [4] Tan,F., Xiong, J., Chaos via Furstenberg family couple, Topology and its Applications , 156(3), (2009) 525–532. [5] Xiong, J. C., L¨u,J., Furstenberg family and chaos, Science in China Series A: Mathematics , 50(9) (2007) 1325–1333. [6] H.Wang , J.Xiong , F.Tan, Furstenberg families and sensitivity, Discrete Dynamics in Nature and Society 2010, (2010).

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Wednesday, June 23, 12:30-12:55 Nikolai Edeko Universit¨atZ¨urich, Switzerland [email protected] A dynamical proof of the van der Corput inequality

For a bounded sequence (un)n∈N in a Hilbert space H, the van der Corput inequality

N 2 J N 1 X 1 X 1 X lim sup u ≤ lim sup lim sup (u |u ) N n J N n n+j N→∞ n=1 J→∞ j=1 N→∞ n=1

9 states that asymptotically, norms of averages can be bounded by averages of correlations. This idea is routinely used for complexity reduction to prove different equidistribution and ergodic theorems. On the other hand, the Furstenberg correspondence principle provides a universal technique to study the asymptotic behavior of scalar sequences in terms of measure-preserving dynamical systems. Can the van der Corput inequality and related asymptotic inequalities thus be reduced to dynamical principles? We will address this question in the scalar case and then discuss how to overcome the limitation of the Furstenberg correspondence principle to scalar sequences. This is joint workwith H. Kreidler and R. Nagel.

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Tuesday, June 22, 12:30-12:55 Antongiulio Fornasiero Universit`adegli Studi di Firenze, Italy [email protected] Algebraic entropy and Hilbert polynomials

Given a module A over S := Z[X1, ..., Xk], people have defined the ”algebraic entropy” ent of A (starting with the case when k = 1), taking inspiration from Kolmogorov-Sinai entropy of self-maps of probability spaces. We show that, to every finite set of generators V of A (as S-module), one can associate a Hilbert polynomial q (using similar construction to the classical case when R is a field): the leading monomial of q is independent from the choice of V , and gives an invariant µ(A). This invariant can be extended to the case when A is not finitely generated, and refines the algebraic entropy. As in the case of ent, µ is additive on exact sequences of torsion groups, and can be extended (keeping additivity) to a functionµ ˜ on all S-modules. The invariant µe is the refinement of the ”intrinsic algebraic entropy” entf . From µ, one can also compute the so-called ”receptive algebraic entropy” and its ”intrinsic” version, and easily deduce the latter is additive. All the above can be extended to algebraic entropy of length functions over Noetherian rings.

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day, June 2, 1:0-1:0 Anna Giordano Bruno Universit`adegli Studi di Udine, Italy [email protected] Receptive topological and metric entropy

A notion of topological entropy, that was later called receptive, was introduced by Gys, Langevin and Walczak [5] (later studied by Bi´s[2]) and independently by Hofmann and Stoyanov [6]. The topic of this talk, based on a joint paper with Bi´s,Dikranjan and Stoyanov [3], is the receptive topological entropy for uniformly continuous monoid actions on metric spaces. We propose several characterizations of this invariant, following the ideas by Adler, Konheim and McAndrew [1], Bowen [4] and Pesin [7]. In analogy with the receptive topological entropy, we introduce the receptive metric entropy for measure preserving monoid actions on probability spaces and study its basic properties. In the case of continuous monoid actions on compact metric spaces admitting an invariant Borel probability measure, we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. [1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. 114 (1965) 309–319. [2] A. Bi´s, Entropies of a semigroup of maps, Discr. Cont. Dyn. Sys. 11 (2004), 639–648. [3] A. Bi´s,D. Dikranjan, A. Giordano Bruno, L. Stoyanov, Metric versus topological receptive entropy of semigroup actions, Qual. Theory Dyn. Syst. DOI:10.1007/s12346-021-00485-7. [4] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 154 (1973), 125–136. [5] E. Ghys, R. Langevin, P. Walczak, Entropie geometrique des feuilletages, Acta Math. 160 (1988), 105–142. [6] K.-H. Hofmann, L. N. Stoyanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995), 54–98. [7] Ya. Pesin, Dimension theory in dynamical systems: contemporary views and applications, The University of Chicago Press, Chicago 1997.

10 Wednesday, June 23, 10:00-10:50 Eli Glasner Tel Aviv University, Israel [email protected] Topological characteristic factors and nilsystems

We prove that the maximal infinite step pro-nilfactor X∞ of a minimal dynamical system (X,T ) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of π : X → X∞ ∗ ∗ ∗ ∗ the induced open extension π : X → X∞ has the following property: for x in a dense Gδ set of X , the orbit −1 2 d
∗ (d)  ∗ (d) ∗ (d) closure Lx = O (x, x, . . . x),T × T × ... × T is (π ) –saturated, i.e. Lx = (π ) (π ) (Lx). Using results derived from the above fact, we are able to answer several open questions: (1) if (X,T k) is minimal for some k ≥ 2, then for any d ∈ N and any 0 ≤ j < k there is a sequence {ni} of Z with ni ≡ j( mod k) such that ni 2ni dni T x → x, T x → x, . . . , T x → x for x in a dense Gδ subset of X; (2) if (X,T ) is totally minimal, then n n2 o T x : n ∈ Z is dense in X for x in a dense Gδ subset of X; (3) for any d ∈ N and any minimal t.d.s., which is open extension of its maximal distal factor, RP[d] = AP[d], where the latter is the regionally proximal relation of order d along arithmetic progressions. This is a joint work with Wen Huang, Song Shao, Benjamin Weiss and Xiangdong Ye.

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Friday, June 25, 12:30-12:55 Oleg Gutik Ivan Franko National University of Lviv, Ukraine [email protected] On some generalization of the bicyclic monoid

F We introduce algebraic extensions Bω of the bicyclic monoid for an arbitrary ω-closed family F subsets of ω which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse F semigroups. It is proven that Bω is combinatorial inverse semigroup and Green’s relations, the natural partial order F on Bω and its set of idempotents are described. We prove the criteria of simplicity, 0-simplicity, bisimplicity, 0- F F bisimplicity of the semigroup Bω . We gave the criteria when the semigroup Bω has the identity, and when the F semigroup Bω is isomorphic to the bicyclic semigroup or the countable semigroup of matrix units. Joint work with Mykola Mykhalenych.

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Thursday, June 24, 11:30-11:55 Till Hauser Friedrich-Schiller-Universit¨at,Germany [email protected] Entropy beyond actions of uniform lattices

Measure theoretical entropy is an intensely studied concept with various applications and interpretations. For actions of non-discrete groups, such as Rd it can be defined by computing the entropy with respect to a uniform lattice, such as Zd. Nevertheless, there exist (metrizable and σ-compact) locally compact Abelian groups, such as the additive group of p-adic numbers, that do not contain uniform lattices. In this talk we explore two non-equivalent notions of entropy, which both generalize the notion of entropy from the setting of discrete amenable groups to the setting of unimodular amenable groups. The first concept is defined by using the concept of (thin) Følner nets from

11 [1]. The second concept will be defined by replacing the uniform lattice by a weaker structure, called a Delone set, which exists in every unimodular amenable group. This concept generalizes the concept considered in [2]. We relate these notions to the respective notions of topological pressure, present a link to naive entropy and proof respective versions of Goodwyn’s half of the variational principle. Joint work with Friedrich Martin Schneider. [1] F. M. Schneider and A. Thom: On Følner sets in topological groups, Compos. Math., 154(7):1333–1361, 2018. [2] AT Tagi-Zade: Variational characterization of topological entropy of continuous transformation groups. case of actions of Rn, Mathematical Notes of the Academy of Sciences of the USSR, 49(3):305–311, 1991.

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Thursday, June 24, 10:30-10:55 Johannes Jaerisch Nagya University, Japan [email protected] Hausdorff dimension of escaping sets for Z-extensions of expanding interval maps

This talk is based on a joint work with M. Gr¨ogerand M. Kesseb¨ohmer[?]. The aim is to investigate transient behavior of expanding interval maps on the real line by means of fractal geometry. Our motivation stems from the dynamics of Fuchsian groups G acting on hyperbolic space H, and the geodesic flow on the unit tangent space of H/G. The recurrent part of the geodesic flow is given by conical limit set Λc(G) ⊂ ∂H, which is a subset of the limit set Λ(G). The transient behavior is reflected by the complement Λ(G) \ Λc(G). It is well known by a result of Bishop and Jones [?] that the Hausdorff dimension of Λc(G) is equal to the Poincar´eexponent of G    X −s·d(0,g0)  δG := inf s ≥ 0 : e < ∞ ,  g∈G  where d denotes the hyperbolic metric on H. If G is finitely generated, then Λ(G) \ Λc(G) is countable. For a normal subgroup N < G we have that dimH (Λc (N)) = dimH (Λ(N)) if and only if G/N is amenable by results of Brooks [?] and Stadlbauer [?]. In this talk we consider expanding interval maps on the real line which can be considered as models for the geodesic flow on the covering surface associated with a normal subgroup N < G such that G/N = Z. We will introduce escaping sets, and use thermodynamic formalism to determine their Hausdorff dimension spectrum. Let F be an expanding interval map with finitely many C1+ full branches, say F | : I → [0, 1] with disjoint Ii i −1 intervals Ii ⊂ [0, 1] with non-empty interior, i ∈ I := {1, . . . , m}, m ≥ 2. Set hi := Hi : [0, 1] → Ii for the continuous N continuation Hi of F |I to Ii and define the corresponding coding map π : I → [0, 1] by π (ω1, ω2,...) := x for T i
hω ◦ · · · ◦ hω ([0, 1]) = {x}. The repeller of F is then π IN ⊂ [0, 1] and we set R := π (Σ) + . We assume n∈N 1 n Z that the step length function Ψ : [0, 1] → Z is constant on each of the intervals Ii and consider the Ψ-lift of F given by X F Ψ : R → R : x 7→ (k + F (x − k) + Ψ(x − k)) 1[0,1](x − k). k∈Z

Let σ : IN → IN denote the left shift and denote symbolic step length function ψ : IN → Z which is constant on one-cylinder sets such that ψ = Ψ ◦ π. The map F Ψ is conjugate to the Z-extension of σ given by

σ o ψ :Σ × Z → Σ × Z, (σ o ψ)(ω, x) := (σ(ω), x + ψ(ω)).

For α ∈ R let us now define the α-escaping set for F Ψ by

n E(α) := {x ∈ R | ∃K > 0 |F Ψ(x) − nα| ≤ K for infinitely many n ∈ N} , and the uniformly α-escaping set for F Ψ by

n Eu(α) : = {x ∈ R | ∃K > 0 ∀n ∈ N |F Ψ(x) − nα| ≤ K} .

12 Define the geometric potential ϕ : IN → (−∞, 0), ϕ (ω) := − log |F 0 (π (ω))| and the α-Poincar´eexponent     X s·Sω ϕ δα := inf s ≥ 0 | e < ∞ , ?  ω∈I :|Sω (ψ−α)|

? S n Pn−1 i where I = n≥1 I , K > 0 is an arbitrary positive number, and Sω1...ωn f = supτ∈IN i=0 f ◦ σ (ω1 . . . ωnτ). Theorem 1. dimH (E(α)) = dimH (Eu(α)) = δα.

Let δ = dimH π(IN) and let µδ denote the equilibrium state for δϕ. R Theorem 2. We have δα = δ if and only if ψ dµδ = α.

Moreover, we show that the α-Poincar´eseries diverges at δα. For α = 0, Theorem 2 can be used to reprove the following statement for Fuchsian groups [4]. For a convex cocompact Fuchsian group G and a normal subgroup N < G such that G/N = Z, the normal subgroup N is of divergence type and δN = δG. Indeed, in this setting one can show R that ψ dµδ = 0 where µδ corresponds to a version of the Patterson-Sullivan measure of dimension δG.

[1] C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179(1): 1–39, 1997. [2] R. Brooks, The bottom of the spectrum of a Riemannian covering, J. Reine Angew. Math. 357: 101–114, 1985. [3] M. Gr¨oger,J. Jaerisch, and M. Kesseb¨ohmer, Thermodynamic formalism for transient dynamics on the real line, arXiv:1905.09077, 2019. [4] M. Rees Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergod. Theory Dyn. Syst. 1(1): 107–133, 1981. [5] M. Stadlbauer, An extension of Kesten’s criterion for amenability to topological Markov chains, Adv. Math. 235: 450–468, 2013.

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Wednesday, June 23, 17:00-17:50 Vadim Kaimanovich University of Ottawa, Canada [email protected] Coincidence, equivalence and singularity of harmonic measures

In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures ”invariant on average”, with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of harmonic measures on the underlying step distributions on the group. The talk is based on joint work with Behrang Forghani.

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13 Wednesday, June 23, 18:00-18:25 Henrik Kreidler Bergische Universit¨atWuppertal, Germany [email protected] The Furstenberg–Zimmer structure theorem revisited

The Furstenberg-Zimmer theorem is a key structural result in ergodic theory. It allows, e.g., to show multiple recurrence for measure-preserving systems which, by Furstenberg’s correspondence principle, is equivalent to the theorem of Szemerdi on artithmetic progressions. In this talk we propose a new operator theoretic approach to this classical result. We show that, in essence, the Furstenberg-Zimmer theorem is a consequence of operator theory on so-called Hilbert-Kaplansky modules which are natural ”relative versions” of classical Hilbert spaces. This functional analytic perspective provides a systematic approach to extensions of measure-preserving systems. In addition, it allows to drop any countability assumptions yielding the structure theorem for actions of arbitrary groups on arbitrary probability spaces. This is a joint work with Nikolai Edeko (Zurich) and Markus Haase (Kiel).

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Tuesday, June 22, 17:00-17:50 Eugen Mihailescu Academia Romˆanˆa,Romania [email protected] Pressure functional in semigroup dynamics

In this talk we will present first some semigroups given by iterated function systems with overlaps. In this setting we study a class of measures on the limit set Λ, and find a geometric formula for their pointwise dimension by using the overlap numbers. Then, we study finitely generated semigroups G on a compact metric space X and several notions of topological pressure and their capacities on non-compact sets Y . This is done in part with Carath´eodory- Pesin structures. In the dynamics of G we prove a Partial Variational Principle for the amalgamated pressure. Local amalgamated entropies and local exhaustive entropies are also studied for measures on X and bounds are obtained in the case of marginal ergodic measures. The amalgamated pressure is then applied to estimate the Hausdorff dimension for certain fractal intersections.

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Thursday, June 24, 12:30-12:55 Yuto Nakajima Kyoto University, Japan [email protected] The Hausdorff dimension of some planar sets with unbounded digits

We consider the following parameterized planar sets with unbounded digits, which are closely related to iterated function systems.

 ∞  X j  ajλ ∈ C : for each j ∈ {0, 1, 2...}, aj ∈ {0, pj} . j=0 

∞ Here, the parameters λ are complex numbers in {λ ∈ C : 0 < |λ| < 1} and the real numbers {pj}j=0 satisfy the following conditions:

14 • For each j ∈ {0, 1, 2, ...}, pj ≥ 1;

• pj → ∞ as j → ∞; • pj+1 → 1 as j → ∞. pj We investigate these sets by using the method of “transversality”, which is the main tool in investigating self-similar sets with overlaps. We calculate the Hausdorff dimension of these sets for typical parameters in some region with respect to the 2-dimensional Lebesgue measure. In addition, we estimate the local dimension of the exceptional set of parameters.

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Tuesday, June 22, 18:00-18:25 Ryszard Pawlak UniwersytetL´odzki,Poland [email protected] On points of distributive chaos

The main aim of the lecture will be to discuss the problem of focusing distributive chaos around a certain point. This issue is the part of the general thematic line considered in our team: the study of local aspects of discrete dynamical system. Our considerations will be connected with dynamical systems created by functions or multifunctions operating in topological manifolds being compact metric space. Following the definitions in [19] and [2] we adopt the notions of lower- and upper distribution function and distributionally scrambled set and distributionally chaotic system (in the case of multifunction, we consider Hausdorff metric). Let (f1,∞) be a dynamical system consisting of functions or multifunctions. We shall say that x0 ∈ X is a DC1 point (distributionally chaotic point) of (f1,∞) if for any ε > 0 there exists an uncountable set S being a DS-set for (f1,∞) such that there are n ∈ N and a closed set A ⊃ S fulfilling the condition

i·n A ⊂ ζ1 (A) ⊂ B(x0, ε) for i ∈ N. The lecture will cover the following issues: • the existence of DC1 points (connected with functions); • approximation of dynamical systems consisting of continuous functions by systems having a DC1 point; • approximating a function by semigroups composed of uniformly 0-approximately continuous functions (we shall say that a family of functions S is uniformly 0-approximately continuous at a point x0 if there exists a Lebesgue measurable set A such that x0 is density point of A, with respect to Lebesgue measure, and lim f(x) = f(x0) A3x→x0 and entropy h(f, A) = 0, for each f ∈ S.) such that each function belonging to the semigroup has a ”big” set of DC1 points which are its discontinuity points; • application of DC1 points of multifunctions to the theory of infinite topological games.

[1] Alsed´aL., Llibre J., Misiurewicz M., Combinatorial Dynamics and Entropy in Dimension One, World Sci., (1993). [2] Balibrea F., Sm´ıtalJ., Stef´ankov´aM.,ˇ The three versions of distributional chaos, Chaos Solitons Fractals 23 (5), 2005, 1581–1583. [3] Beer G., The approximation of upper semicontinuous multifunction by step multifunction, Pacific Journal of Mathematics 87. 1 (1980), 11–19. [4] Ciklov´aM.,ˇ Dynamical systems generated by functions with connected Gδ graphs, Real Analysis Exchange, vol. 30(2), 2004/2005, 617–638. [5] Hou B., Wang X. Entropy of a semigroup of maps from a set-valued view, Bull. Iranian Math. Soc. Vol. 43 (2017), No. 6, pp. 18211835. [6] Krom M. R., Infinite games and special Baire space extensions, Pacific J. Math., 55 (2), 1974, 483–487.

15 [7] Illanes A., Nadler Jr SB. Hyperspaces, Monographs and textbooks in pure and applied mathematics, vol. 216. New York: Marcel Dekker Inc. (1999). [8] Komenskii M., Obukhovskii V., Zecca P. Condensing multivalued maps and semilinear differential inclusions in Banach spaces, Walter de Gruyter, Berlin New York (2001). [9] Korczak-Kubiak E., Loranty A., Pawlak R. J., Baire generalized topological spaces, generalized metric spaces and infinite games, Acta Math. Hungar. 140 (3) (2013), 203231. [10] Kwietniak D., Oprocha P. Topological entropy and chaos for maps induced on hyperspaces, Chaos, Solitons and Fractals 33 (2007), 7686. [11] Lee, J. M. [2000] Introduction to Topological Manifolds’, Graduate Texts in Mathematics (Springer-Verlag). [12] Michael E. Continuous selections I, Ann. Math. 63 (1956), 361382. [13] Oprocha P., Distributional chaos revisited, Trans. Amer. Math. Soc. 361 (9), 2009, 4901–4925. [14] Pawlak R. J., Loranty A., The generalized entropy in the generalized topological spaces, Topology and its Applications 159 (2012) 17341742. [15] Pawlak R. J., Loranty A., On the local aspects of distributional chaos, Chaos 29, 013104 (2019). [16] Pawlak R. J., Distortion of Dynamical Systems in the Context of Focusing the Chaos Around the Point, International Journal of Bifurcation and Chaos, Vol. 28, No. 1 (2018) (13 p.). [17] Ruette S. Chaos on the interval a survey of relationship between the various kinds of chaos for continuous interval maps, arXiv:1504.03001 [18] Schweizer B., Sklar A., Sm´ıtalJ., Distributional (and other) chaos and its measurement, Real. Anal. Exch. 26 (2), 2000, 495–524. [19] Schweizer B., Sm´ıtalJ., Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (2), 1994, 737–754. [20] Telg´arskyR., Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math., 17 (2), 1987, 227–276. [21] Ye X., Zhang G., Entropy points and applications, Trans. Amer. Math. Soc. 359.12, 2007, 61676186.

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Tuesday, June 22, 10:00-10:50 Mark Pollicott University of Warwick, UK [email protected] Rate of escape for random walks on SL(2, R)

Given a finitely supported measure on SL(2, R) one can consider the associated random walk, and the associated rate of escape, or drift. We will consider the problem of estimating this numerical value, which has applications to determining whether the associated hitting measure is singular. This is joint work with P. Vytnova.

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Friday, June 25, 16:30-17:20 Feliks Przytycki Instytut Matematyczny Polskiej Akademii Nauk, Poland [email protected] On Hausdorff dimension of polynomial not totally disconnected Julia sets

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case (Zdunik, 1990, 1991). We give also an (easy) example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs This is a joint work with Anna Zdunik, to be published in Bull. LMS.

16 Tuesday, June 22, 11:00-11:50 MichalRams Instytut Matematyczny Polskiej Akademii Nauk, Poland [email protected] Restricted variational principle for partially hyperbolic dynamics

The variational principle states that the topological entropy of a compact dynamical system is a maximum of metric entropies of invariant measures supported on this system. If the system is not compact then one direction of the inequality still holds: there cannot exist an invariant measure with metric entropy larger than the topological entropy of the system. It is a classical and very useful result. In the multifractal formalism we consider the division of a (smooth) dynamical system into subsets of points with given Lyapunov exponent. Those subsets are invariant but usually not compact. In several simple cases it was then possible to prove the so-called restricted variational principle: that for every possible value α of the Lyapunov exponent the topological entropy of the set of points with exponent α is equal to the supremum of metric entropies of ergodic measures with Lyapunov exponent α. In particular, in a paper with Lorenzo Diaz and Katrin Gelfert we prove the restricted variational principle for a class of skew-products (basically, action of a free group of circle diffeomorphisms) for al α 6= 0. In general, it is much more difficult to construct ergodic measures with zero exponent. For cocycles of circle diffeomorphisms, the first construction of an zero exponent ergodic measure was given by Gorodetski, Ilyashenko, Kleptsyn, Nalski, the measure they constructed had always zero entropy. Bochi, Bonatti, Diaz gave then a construction of a positive entropy ergodic measure with zero exponent. The result I will present is the construction of a sequence of ergodic zero exponent measures with metric entropies converging to the maximal possible value (the topological entropy of the set of points with zero Lyapunov exponent). It was done together with Lorenzo Diaz and Katrin Gelfert, and together with our previous result it gives the full restricted variational principle for our class of skew products, which is open and dense in the class of all cocycles of circle diffeomorphisms.

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Wednesday, June 23, 18:30-18:55 Konstantin Slutsky Iowa State University, USA [email protected] L1 full groups of measure-preserving actions of Polish normed groups

We introduce the concept of an L1 full group associated with a Borel measure-preserving action of a Polish normed group, which extends the previously studied case of actions of discrete groups. We show that these groups carry a natural Polish topology and that they are topologically simple whenever the action is ergodic. Structure of L1 full groups is best understood through the study of their topological derived subgroups. When the acting group is locally compact and amenable, this derived subgroup is whirly amenable and (under minor assumptions) topologically 2-generated. In the case of flows—that is, actions of R—we identify the derived subgroup with the kernel of the index map. Orbit dynamics of elements in full groups of locally compact group actions differs from that of the discrete groups due to the non-trivial conservative part in the Hopf decomposition. In this talk we will discuss the new phenomena this brings to the structure of L1 full groups. Joint work with Fran¸cois Le Maˆıtre.

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17 Wednesday, June 23, 13:00-13:25 Lubom´ırSnoha´ Univerzita Mateja Bela, Slovakia [email protected] Product of minimal spaces: complete solution

This is a joint work with M. Dirb´akand V. Spitalsk´y.ˇ A metric space is said to be a minimal space, if it admits a minimal (not necessarily invertible) map; for some references see e.g. [6]. Even such a basic and natural question, explicitly posed in [2 p. 126], as whether the product of two compact minimal spaces is a minimal space, has not been answered so far in its full generality, though recently two results related to the question have appeared: • The class of compact metric spaces admitting minimal continuous flows is closed with respect to at most countable products, see [3, Theorem 25]. • In the special case when homeomorphisms rather than continuous maps are considered, a negative answer has been provided. There is a metric continuum Y admitting a minimal homeomorphism such that Y × Y does not admit any minimal homeomorphism [1] (the space Y is obtained as a modification of a space from [5]. We solve the problem completely:

• The answer to the question is negative, i.e. the product of minimal spaces need not be minimal [4, Theorem A]. In fact there exists a continuum such that it admits both a minimal homeomorphism and a minimal noninvertible map, but its square admits neither a minimal homeomorphism nor a minimal noninvertible map.

[1] J. P. Boro´nski,A. Clark, P. Oprocha, A compact minimal space Y such that its square Y × Y is not minimal, Adv. Math. 335 (2018), 261–275. [2] H. Bruin, S. Kolyada, L’. Snoha, Minimal nonhomogeneous continua, Colloq. Math. 95 (2003), no. 1, 123–132. [3] M. Dirb´ak, Minimal extensions of flows with amenable acting groups, Israel J. Math. 207 (2015), no. 2, 581–615. [4] M. Dirb´ak,L’. Snoha, V. Spitalsk´y,ˇ Minimal direct products, submitted; see arXiv:2005.06969 [math.DS]. [5] T. Downarowicz, L’. Snoha, D. Tywoniuk, Minimal Spaces with Cyclic Group of Homeomorphisms, J. Dy- nam. Differential Equations 29 (2017), no. 1, 243–257. [6] S. Kolyada, L’. Snoha, Minimal dynamical systems, Scholarpedia 4(11):5803 (2009), http://www.scholarpedia. org/article/Minimal_dynamical_systems.

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Friday, June 25, 13:00-13:25 Adam Spiewak´ Bar-Ilan University, Israel [email protected] Singular stationary measures for random piecewise affine interval homeomorphisms

We consider a family of iterated function systems, each consisting of two (symmetric) piecewise affine increasing homeomorphisms of the unit interval, each with exactly one point of non-differentiability. We call them Alsed´a- Misiurewicz systems (as they were introduced and studied by Alsed´aand Misiurewicz). Under certain assumptions, such a system admits a unique stationary probability measure with no atoms at the endpoints. It has to be either singular or absolutely continuous with respect to the Lebesgue measure. Alsed´aand Misiurewicz conjectured that typically such measures should be singular. We prove that singularity holds for a certain open set of parameters, as well as systems satisfying some resonance conditions. In the latter case, we calculate or bound Hausdorff dimension of the stationary measure and its support. This is a joint work with Krzysztof Bara´nski.

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18 Thursday, June 24, 13:00–13:50 Manuel Stadlbauer Universidade Federal do Rio de Janeiro, Brazil [email protected] Quenched and annealed thermodynamic formalism for semigroups

The joint action of a finite number T1,...,Tk of Ruelle expanding maps might be studied by means of thermody- namic formalism as this approach, among other tools, comes with a natural selection principle through the variational principle. In order to so, we fix Hlder continuous potentials ϕj (1 ≤ j ≤ n) and consider the semigroup S generated by {(Tj, ϕj) : 1 ≤ j ≤ n} with respect to the action

(Tj, ϕj)(Ti, ϕi) = (Tj ◦ Ti, ϕj ◦ Tj + ϕi).

In this expanding setting, each g = (T, ϕ) ∈ S comes with a unique equilibrium state µg which might studied in detail by the associated Ruelle operator, which is defined by, for f : X → [0, ∞),

X ϕ(y) Lg(f)(x) := e f(y). T y=x

In order to study the thermodynamic formalism of S, we will discuss the quenched and the annealed approach. The quenched approach analyses the evolution of µg as g tends to infinity. That is, by defining a distance between g and h through the Wasserstein distance of µg and µh, one obtains a compactification of S which identifies trajectories which have asymptotically the same equilibrium states. Furthermore, the associated boundary ∂S recovers algebraic properties of S. For example, ∂S is trivial whenever S is abelian. On the other, there are examples of free semigroups where ∂S is isomorphic to {0, 1}Z. The annealed approach, on the other hand, studies S from the point of view of a random walk. That is, by fixing a probability measure P a probability measure on {1, . . . , k}N, the annealed operator Z

An := Ljn ◦ · · · ◦ Lj1 dP (j1 . . . jn) describes the average action of the semigroup on functions at time n. If P satisfies the Gibbs-Markov condition, it then follows that An(f)/An(1) converges exponentially fast which gives rise to the annealed equilibrium state. In addition, this annealed state can be written as an integral of the elements of ∂S from the quenched approach. This is joint work with Paulo Varandas and Xuan Zhang. The preprint can be found here: https://arxiv.org/abs/2004.04763

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Friday, June 25, 10:00-10:50 Hiroki Sumi Kyoto University, Japan [email protected] Random dynamical systems of regular polynomial maps on C2

We consider i.i.d. random dynamical systems of regular polynomial maps on C2 (2-dimensional complex Euclidean space) and P2 (2-dimensional complex projective space). We show that for a generic such system which has an attractor in the line at infinity, the system has the “mean stability”, i.e., (1) there exist finitely many attractors A1,...,Am of the systems and for each initial value z in P2, there exists a random orbit of z with respect to the system which tends 2 to one of the attractors A1,...,Am, and (2) for each initial value z in P , for almost every sequence of the maps in the system, the maximal Lyapunov exponent of the sequence at z is less than a negative constant c, where c depends only on the system, and does not depend on the initial value z and the sequence. Note that the above things are new phenomena in random holomorphic dynamical systems which cannot hold for deterministic iteration dynamics of a single holomorphic map of degree two or more. We see that randomness or

19 noise can easily give us many interesting new phenomena in random dynamical systems. Such phenomena are called “randomness-induced phenomena” or “noise-induced phenomena”. In this talk, we focus on “randomness-induced order” (or “noise-induced order”) in random dynamical systems. When we deal with i.i.d. random holomorphic dynamical systems, it is very important to consider the dynamics of associated semigroups G of holomorphic maps. In fact, we consider the Fatou set F (G) of G (the maximal open set in P2 where G is equicontinuous) and the Julia set J(G) := P2 \ F (G) of G. Also, we consider the minimal sets of G. 2 Here, a non-empty compact subset K of P is called a minimal set of G if for each z ∈ K, we have ∪h∈G{h(z)} = K. For the papers of 1-dimensional random holomorphic dynamical systems which deal with randomness-induced phenomena, see the reference list. [1] H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc., (2011), 102 (1), 50–112. [2] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, Adv. Math. 245 (2013) 137–181. [3] H. Sumi, Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Sys- tems and Random Relaxed Newton’s Methods, to appear in Comm. Phys. Math. https://arxiv.org/abs/1608.05230.

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Tuesday, June 22, 13:30-13:55 Andrei Tetenov Novosibirsk State University & Gorno-Altaisk State University, Russia [email protected] Three theorems on self-similar continua with finite intersection property

n Let S = {S1, ..., Sm} be a system of injective contraction maps in R , whose attractor K is connected. If for any non-equal i, j ∈ {1, ..., m}, the intersection of the pieces Ki and Kj of the attractor K contains at most s points, then we say that the system S is a FI(s)-system of contractions and we say that K is a self-similar continuum with finite intersection property. For a FI(s)-system of contractions S we define its bipartite intersection graph Γ(S). This graph determines the topological properties of the attractor K. Particularly, we prove the following Theorem. Theorem 1. Let S be a system of injective contraction maps in a complete metric space X, which satisfies finite intersection property. The attractor K of the system S is a dendrite iff the intersection graph Γ(S) of the system S is a tree. If a FI(s)-system S satisfies OSC, then its attractor possesses various finiteness properties: Theorem 2. If S is a FI(s)-system of similarities in Rn which satisfies open set condition, then there is a finite −1 −1 upper bound for the numbers of addresses #π (x) and #π (∂Kj); for the ramification numbers NC ({x}) and NC (Kj) and for the topological order Ord(x, K) and Ord(Kj,K). Applying these results to FI(s)-systems of similarities in R2 (viewed as C), we define a slope parameter of a Jordan arc. We say that a Jordan arc γ ⊂ C has a slope parameter λ at the endpoint a if there is M > 0 such that for any 0 0 subarc γ = γ(z1, z2) ∈ γ˙ the increment ∆γ0 arg(z − a) of the argument of z − a along γ satisfies the inequality

|∆γ0 arg(z − a) − λ(log |z2 − a| − log |z1 − a|)| ≤ M

We boundary points have a slope parameter property. Namely, the following theorem is true. 2 Theorem 3. If S is a FI(s)-system of similarities in R , and p is a boundary point of a piece Ki which has a preperiodic address, then all Jordan arcs γ ⊂ K with the end point p have a slope parameter λ(x), which is the same for all these arcs.

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20 Thursday, June 24, 18:00-18:25 Mariusz Urba´nski University of North Texas, USA [email protected] Random non–hyperbolic exponential maps

Joint work with Anna Zdunik. We will consider random iteration of exponential entire functions, i.e. functions of the form

z C 3 z 7−→ fλ(z) := λe ∈ C, λ ∈ C \{0}.

Assuming that λ is in a bounded closed interval [A, B] ⊂ R with A > 1/e, I will speak about random iteration of the maps fλ governed by an invertible measurable map

θ :Ω −→ Ω preserving a probability ergodic measure m on Ω, where Ω is a measurable space. The link from Ω to exponential maps is then given by an arbitrary measurable function

η :Ω 7−→ [A, B].

I will in fact deal with the cylinder space Q := C/ ∼, where ∼ is the natural equivalence relation: z ∼ w if and only if w − z is an integral multiple of 2πi. I will discuss the following dynamical results:

(1) For every t > 1 there exists a unique random conformal measure ν(t) for the random conformal dynamical system on Q. (2) The measure ν(t) is supported on the, appropriately defined, radial Julia set. (3) There exists a unique random probability invariant measure µ(t) absolutely continuous with respect to ν(t). (4) The measures µ(t) ν(t) are equivalent.

Turning to geometry I will define an expected topological pressure EP(t) ∈ R and will discuss the following results:

(5) The only zero h of EP coincides with the Hausdorff dimension of m–almost every fiber radial Julia set Jr(ω) ⊂ Q, ω ∈ Ω. (6) h ∈ (1, 2)

(7) The omega–limit set of Lebesgue almost every point in Q is contained in the real line R.

I will also show how to entirely transfer our results to the original random dynamical system on C.

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Thursday, June 24, 17:00-17:50 Tom Ward Newcastle University, UK [email protected] Order of mixing in algebraic systems

I will give an overview of the Diophantine and number-theoretic issues that arise in asking about mixing properties of algebraic actions generated by countably many group automorphisms.

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21 Thursday, June 24, 10:00-10:25 Takayuki Watanabe Kyoto University, Japan [email protected] Non-i.i.d. random holomorphic dynamical systems

We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on “Markovian rules”. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the Lyapunov exponent is uniformly negative for every initial value and almost every random orbit. Also, we show the continuity of the function which represents the probability of random orbits tending to an attracting minimal set. This is joint work with Hiroki Sumi (Kyoto University).

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Friday, June 25, 11:00-11:50 Benjamin Weiss Hebrew University of Jerusalem, Israel [email protected] The Foias-Stratila theorem in higher dimensions

It is a remarkable result of C. Foias and S. Stratila that if the spectral measure of an ergodic stationary process is continuous and supported on K ∪ −K where K is a Kronecker set on the circle then the process must be Gaussian. Recall that K is a Kronecker set if the exponentials are dense in the uniform norm in the group of all continuous functions on K with absolute value one. I will discuss a generalization of this result to stationary processes indexed by Zd.

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22 Participants

1. Jon Aaronson, Tel Aviv University, Israel 2. Jason Atnip, University of New South Wales, Australia 3. Marek Badura, UniwersytetL´odzki,Poland 4. Adam Bartoszek, UniwersytetL´odzki,Poland 5. Vitaly Bergelson Ohio State University USA 6. Sergey Bezuglyi University of Iowa United States 7. Andrzej Bi´s, UniwersytetL´odzki,Poland 8. Everaldo de Mello Bonotto, Universidade de S˜aoPaulo, Brazil 9. Raimundo Brice˜no, PUC – Santiago de Chile, Chile 10. Zolt´anBuczolich, ELTE Etvs Lor´and University, Hungary 11. Alexander Bufetov, Aix-Marseille Universite & CNRS, France 12. Eric Alberto Cabezas Bonilla, UFRJ, Brazil 13. Jose S. C´anovas, Universidad Polit´ecnicade Cartagena, Spain 14. Marisa Cantarino, IMECC – University of Campinas, Brazil 15. Maria Carvalho, Universidad do Porto, Portugal 16. Solly Coles, University of Warwick, UK 17. Maciej Czarnecki, UniwersytetL´odzki,Poland 18. Mauricio D´ıaz, Universidad del B´ıoB´ıo,Chile 19. Dikran Dikranjan, University of Udine, Italy 20. Michal Doucha, Institute of Mathematics, Czech Academy of Sciences Czech Republic 21. Tomasz Downarowicz, Politechnika Wroclawska, Poland 22. Nikolai Edeko, University of Z¨urich, Switzerland 23. Tanja Eisner, University of Leipzig, Germany 24. Antongiulio Fornasiero, University of Florence, Italy 25. Alex Furman, University of Illinois at Chicago, USA 26. Anna Giordano Bruno, University of Udine, Italy 27. Jakub Gismatullin, Uniwersytet Wroc lawski, Poland 28. Eli Glasner, Tel Aviv University, Israel 29. Arek Goetz, San Francisco State Univeristy, USA 30. Cecilia Gonzales Tokman, University of Queensland, Australia 31. Oleg Gutik, National University of Lviv, Ukraine 32. Till Hauser, Friedrich-Schiller-University Jena, Germany

23 33. David Hui, HKUST, Hong Kong 34. Sebastian Hurtado, University of Chicago, USA 35. Johannes Jaerisch, Nagoya University, Japan 36. Sakshi Jain , University of Rome Tor Vergata, Italy 37. Natalia Jurga, University of St Andrews, UK 38. Vadim Kaimanovich, University of Ottawa, Canada 39. Minsung Kim , Uniwersytet Mikolaja Kopernika, Poland 40. Wojciech Koz lowski, UniwersytetL´odzki,Poland 41. Henrik Kreidler, University of Wuppertal, Germany 42. Dominik Kwietniak, Uniwersytet Jagiello´nski,Poland 43. Marco Lopez, Wake Forest University, USA 44. Anna Loranty, UniwersytetL´odzki,Poland 45. Dongkui Ma, South China University of Technology, China 46. Rodica Marineac, Institute of Mathematics of the Romanian Academy, Romania 47. Eugen Mihailescu, Romanian Academy, Romania 48. Andreas Mountakis, University of Warwick, UK 49. Rainer Nagel, University Tuebingen, Germany 50. Yuto Nakajima, Kyoto University, Japan 51. Leticia Pardo Simon, Instytut Matematyczny Polskiej Akademii Nauk, Poland 52. Nicol´oPaviato, University of Warwick, UK 53. Ryszard J. Pawlak, UniwersytetL´odzki,Poland 54. Sebastian A. P´erez, Pontificia Universidad Cat´olicade Valpara´ıso,Chile 55. Mark Pollicott, University of Warwick, UK 56. Feliks Przytycki, Instytut Matematyczny Polskiej Akademii Nauk, Poland 57. Margarita Quispe Tusco, Universidad La Frontera, Chile 58. Micha lRams, Instytut Matematyczny Polskiej Akademii Nauk, Poland 59. Colin Reid, University of Newcastle, Australia 60. Juan Carlos Salcedo, Universidade Federal do Rio de Janeiro, Brazil 61. Shrey Sanadhya, University of Iowa, USA 62. Cagri Sert, University of Z¨urich, Switzerland 63. Richard Sharp, University of Warwick, UK 64. Ruxi Shi, Instytut Matematyczny Polskiej Akademii Nauk, Poland 65. Yi Shi, Peking University, China 66. Jaqueline Siqueira, Federal University of Rio de Janeiro, Brazil 67. Konstantin Slutsky, Iowa State University, USA 68. Lubom´ırSnoha´ , Matej Bel University, Slovakia 69. Boris Solomyak, Bar-Ilan University, Israel 70. Adam Spiewak´ , Bar-Ilan University Israel 71. Manuel Stadlbauer, Universidade Federal do Rio de Janeiro, Brazil

24 72. Luchezar Stoyanov, University of Western Australia, Australia 73. Hiroki Sumi, Kyoto University, Japan 74. Peng Sun, Central University of Finance and Economics, China 75. Andrei Tetenov, Novosibirsk State university & Gorno-Altaisk State University, Russia 76. Jakub Tomaszewski, Uniwersytet Jagiello´nski,Poland 77. Andrew T¨or¨ok, University of Houston, USA 78. Alexandre Trilles, Uniwersytet Jagiello´nski,Poland 79. Sascha Troscheit, University of Vienna, Austria 80. Serge Troubetzkoy, Aix-Marseille University, France 81. Takashi Tsuboi, Musashino University, Japan 82. Siming Tu Sun, Yat-sen University, China 83. Mariusz Urba´nski, University of North Texas, USA 84. Sandro Vaienti, University of Toulin & Center of Theoretical Physics Marseille, France 85. Paulo Varandas, FCT - CMUP & Federal University of Bahia, Portugal 86. Victor Vargas, National University of Colombia, Colombia 87. Helder Vilarinho, Universidade da Beira Interior, Portugal 88. Helmuth Villavicencio Fern´andez, Instituto de Matem´aticay Ciencias Afines, Peru 89. Pawe lWalczak, UniwersytetL´odzki,Poland 90. Tom Ward, Newcastle University, UK 91. Takayuki Watanabe, Kyoto University, Japan 92. Benjamin Weiss, Hebrew University of Jerusalem, Israel 93. Christian Wolf, The City College of New York , USA

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