**Organizers** Katrin Gelfert, Anna Zdunik

**Schedule**

- Thursday
- 09:30
*Lorenzo J. Díaz**Attracting graphs of skew products with non-contracting fiber maps* - 10:30
*Michal Rams**Lyapunov spectrum of GL(2,R) matrix cocycles* - 11:30
*Tomasz Szarek**Ergodicity of random homeomorphisms on the circle* - Lunch
- 14:00
*Klaudiusz Czudek**The rate of convergence for iterated function systems on the circle* - 14:30
*Łukasz Pawelec**Hausdorff dimension for nonautonomous exponential maps* - 15:00
*Adam Śpiewak**Topological Wiener-Wintner theorem for semigroups of Markov operators* - Café
- 16:00
*Dominik Kwietniak**Amorphic complexity beyond Z-actions*

- 09:30
- Friday
- 09:30
*Katrin Gelfert**Random iterations of homeomorphisms on the circle* - 10:30
*Edgar Matias**Synchronization of Markovian random products of homeomorphisms on the circle* - 11:30
*Anna Zdunik**Random conformal dynamical systems and random invariant measures* - Lunch

- 09:30

**Abstracts**

*Lorenzo J. Díaz**Abstract: We study attracting graphs of step skew products from the topological and ergodic points of view where
the usual contracting-like assumptions of the fiber dynamics are replaced by weaker merely topological conditions. In this context, we
prove the existence of an attracting invariant graph and study its topological properties. We prove the existence of globally attracting measures and we show that (in some specific cases) the rate of convergence to these measures is
exponential. (in collaboration with E. Mathias)*

*Michal Rams**Abstract: The affine iterated function system on the plane is a set of k contracting maps of the form f_i(x) = A_i x + a_i, where A_i in GL(2,R) are the linear parts and a_i in R^2 are the translations. Strong regularity means that there doesn't exist a finite set of
directions which is preserved by all the linear parts A_i. Strong separation condition means
existence of an open set U in R^2 such that the sets f_i(U) are contained in U and pairwise disjoint. Given omega in {1,…,k}^N we denote pi(omega) = lim f_{omega_n}…f_{omega_1}(0). We also denote by lambda_1(omega), lambda_2(omega) the Lyapunov exponents (in the Oseledets sense) of the infinite product of matrices A_{omega_1}… A_{omega_n}…. For the set L_{alpha_1,alpha_2} = {omega; lambda_1(omega)=alpha_1, lambda_2(omega) = alpha_2} our goal is to calculate the functions (alpha_1, alpha_2)-> h_{top} L_{alpha_1, alpha_2} and (alpha_1,alpha_2)-> dim_H pi(L_{alpha_1, alpha_2}). This is a joint work with Balazs Barany, Thomas Jordan, and Antti Kaenmaki.*

*Tomasz Szarek**Abstract: We show an elementary proof of ergodicity for iterated function systems on the circle consisting of minimally acting homeomorphisms.*

*Klaudiusz Czudek**Abstract: I will present a proof that, under some assumptions, the Markov operator corresponding to a system of C^1 orientation preserving diffeomorphisms on the circle is stable and the rate of convergence to a unique invariant probability measure is exponential. The proof is provided by V. Kleptsyn.*

*Łukasz Pawelec**Abstract:In this talk we investigate the behaviour of a non-autonomous dynamical system consisting of the exponential maps. In other words, we iterate lambda e^z, where lambda in R_+ — changing the lambda parameter at every step. As in the standard, i.e. autonomous, case we may define the dynamical coding. We study the set of points with the (seemingly most interesting) code (0,0,0…) and prove that its Hausdorff dimension is equal to one.*

*Adam Śpiewak**Abstract: Classical Wiener-Wintner ergodic theorem establishes almost sure convergence of a sequence 1/N sum_{n=0}^{N-1} lambda^n f(T^n x) for every lambda on unit circle, where T is an ergodic measure preserving transformation and f : X -> C is integrable. We prove a generalization of its topological version for semigroups of Markov operators on C(X). We assume that { S_g : g in G } is a mean ergodic representation of a right amenable semitopological semigroup G by linear Markov operators on C(X), where X is a compact Hausdorff space. The main result is necessary and sufficient conditions for mean ergodicity of a distorted semigroup { chi(g)S_g : g in G }, where chi is a semigroup character. Such conditions were obtained before under the additional assumption that { S_g : g in G } is uniquely ergodic. This is joint work with Wojciech Bartoszek.*

*Katrin Gelfert**Abstract: We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.*

*Edgar Matias**Abstract: We study Markovian random products of homeomorphisms on the circle. Recently, Malicet stablished an invariant principle for random products of homeomorphisms on the circle over non-invertible dynamical systems and use it to obtain local synchronization in the i.i.d. case. We will present a version of the invariant principle for random products over invertible dynamical systems and show how to use it to obtain local synchronization in the Markovian case.*

*Anna Zdunik**Abstract: after introducing the notion of a random measure in a Polish space, I will discuss random conformal repellers, random conformal measures and the formula describing the dimension of random conformal repellers. In the context of random iterates of transcendental maps, random Julia and radial Julia sets will be defined. I will present recent results obtained in a collaboration with Mariusz Urba'nski. For a random iteration of non-hyperbolic exponential maps z->lambda exp(z) (within some range of parameters) we prove the existence of random conformal and invariant measures supported on the radial Julia set, and describe the behoviour of the typical trajectory (in the sense of Lebesgue measure).*

*Dominik Kwietniak**Amorphic complexity is a new topological invariant suitable for zero entropy systems. Amorphic complexity distinguishes between minimal mean equicontinuous systems of different complexity. The talk is intended as an introduction to amorphic complexity and a report on an ongoing project with Fuhrmann, Gröger, and Jäger aiming at extension of the amorphic complexity to actions of amenable groups and beyond. Time permits I would indicate applications to cut and project sets (quasicrystals).*