University of Zurich, Rämistrasse 71 (Center campus = Zentrum), at room **KOL-F-121 **(F indicates the floor (e.g. floor A, floor B, etc.), 121 is the room number on the F floor).

Note: This is *not* Irchel campus where UZH Mathematics Department is.

**Monday** ** **

9:00-9:30: Registrations (Lichthof Nord)

9:30-10:30: Frantzikinakis

10:30-11:00: Coffee break

11:00-12:00: Brown

12:00-14:00: Lunch break

14:00-15:00: Quint

15:00-15:45: Coffee break

15:45-16:45: Mohammadi

16:45-17:15: Coffee break

17:15-18:15: Wang

18:45-: Conference Cocktail (Irchel)

**Tuesday **

9:30-10:30: Kleptsyn

10:30-11:00: Coffee break

11:00-12:00: Weiss

12:00-14:00: Lunch break

14:00-15:00: Dolgopyat

15:00-15:45: Coffee break (**Group photo** at 15:00)

15:45-16:45: Zhang

16:45-17:15: Coffee break

17:15-18:15: Dujardin

18:45-: Conference Dinner (Zurichberg Hotel)

**Wednesday** (mind **early** timing) ** **

**9:15-10:15**: Markloff

10:15-10:45: Coffee break

10:45-11:45: Oh

11:45-12:00: Break

12:00-13:00: Hertz

13:00 -: Free afternoon

**Thursday**

9:30-10:30: Lefeuvre

10:30-11:00: Coffee break

11:00-12:00: Butt

12:00-14:00: Lunch break

14:00-15:00: Kra

15:00-15:45: Coffee break

15:45-16:45: Moreira

16:45-17:15: Coffee break

17:15-18:15: Hochman ** **

**Friday**

** **9:30-10:30: Li** **

10:30-11:00: Coffee break

11:00-12:00: Feng

12:00-: Farewell

** **

**Monday**

**Nikos Frantzikinakis:***Ergodic properties of multiplicative functions*

Following the pioneering work of Furstenberg in the 70's, measure preserving systems have been used to model statistical properties of bounded sequences. An in depth understanding of structural properties of these systems often had exciting and quite unexpected implications in combinatorics and number theory. In this talk, I will focus on sequences given by multiplicative functions that take values on the complex unit disc and describe our current understanding of the corresponding measure preserving systems. The new results are based on recent joint work with Mariusz Lemanczyk and Thierry de la Rue.

**Aaron Brown:***Projective actions of higher-rank lattices*

For n at least 3, we aim to understand effective actions of SL(n, Z) on low dimensional manifolds. We are especially interested in actions in dimension (n-1) where we classify all actions as lifts of projective actions and volume-preserving actions in dimension n where we are able to detect projective blow-ups invariant under the action. The main technical tool is a measure rigidity argument for certain non-uniformly hyperbolic actions of higher-rank abelian groups. This is joint project with Federico Rodriguez Hertz and Zhiren Wang.

**Jean-François Quint:***Local limit theorems for conditioned Birkhoff sums*

For a centered random walk $S_n$ on $\mathbb R$, given x in $\mathbb R$, one can estimate the probability that $x+S_n$ belongs to an interval $[a,b]$, while all the $x+S_k$, $1\leq k\leq n-1$, have remained non-negative: it decays as $n^{-3/2}$. Following a general principle which one can date back to Sinai, this should have an analogue for Birkhoff sums of H\"older continuous observables over hyperbolic dynamical systems. Indeed, this is the case, as follows from our joint work with Ion Grama and Hui Xiao.

**Amir Mohammadi:***Dynamics on homogeneous spaces: a quantitative viewpoint*

Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. We will give an overview of activities pertaining to the quantitative aspect of the analysis in this context with an emphasis on recent developments.

**Zhiren Wang:***Fourier decay of self-conformal measures for non-linear IFS’s*

We show that for a C^2 IFS on R, either up to smooth conjugacy the IFS has vanishing second derivative on its attractor, or the self-conformal measure has polynomial decay of Fourier coefficients. A key argument is a cocycle version of Dolgopyat's method and resulting spectral gap-type estimates and renewal theorem. This is a joint work with Amir Algom and Federico Rodriguez Hertz.

**Tuesday**

**Jens Marklof:***Statistics of directions*

Given an infinite point set in R* ^{n}*, say, it is natural to investigate the distribution of directions in which points appear to a fixed observer. Perhaps surprisingly, even strongly correlated sets such as lattices and quasicrystals show intriguing limit distributions: these follow -in some instances- from an application of Ratner’s measure classification theorem for unipotent actions. I will survey some of the key results on directions in both Euclidean and hyperbolic geometry, and explain their connection to fundamental questions in dynamics and number theory.

**Barak Weiss:***Pushforwards of rational fractal measures*

Let nu be a Bernoulli measure on a fractal in R^d generated by a finite collection of contracting similarities with no rotations and with rational coefficients; for instance, the usual coin tossing measure on Cantor's middle thirds set. Let a_t = diag (e^t, …, e^t, e^{-dt}), let U be its expanding horospherical group, which we identify with R^d, and let \bar nu be the pushforward of nu onto the space of lattices SL_{d+1}(R)/SL_{d+1}(Z), via the orbit map of the identity closet under U. In joint work in progress with Khalil and Luethi, we show that the pushforward of \bar mu under a_t equidistributes as t tends to infinity, as do the pushforwards under more general one parameter subgroups. This generalizes a previous result of Khalil and Luethi. I will discuss some Diophantine applications and some ideas used in the proof.

**Dimitry Dolgopyat:***Local Limit Theorems for non stationary systems*

I will review limit theorems for sums of non stationary random variables concentrating on observables for non autonomous dynamical systems and on using dynamical systems methods for proving local limit theorems. The new results presented in the talk are based on joint works with Omri Sarig and with Yeor Hafouta.

**Zhiyuan Zhang:***Newhouse phenomenon in the complex Hénon family*

In a work in progress with Artur Avila and Mikhail Lyubich, we show that there are maps in the complex Hénon family with a stable homoclinic tangency. Moreover, we show that any analytic unfolding of a quadratic homoclinic tangency of a dissipative saddle periodic point of a holomorphic map in \mathbb{C}^2 possesses a parameter with a stable homoclinic tangency. We will explain a new mechanism for the stable intersections between two dynamical Cantor sets generated by two classes of conformal IFSs on the complex plane.

**Romain Dujardin:***Degenerate homoclinic bifurcations in complex dimension 2*

Unfolding homoclinic tangencies is the main source of bifurcations in 2-dimensional (real or complex dynamics). When studying this phenomenon, it is common to assume that tangencies are quadratic and unfold with positive speed. Adapting to the complex setting an argument of Takens, we show that any 1-parameter family of 2-dimensional diffeomorphisms unfolding an arbitrary tangency contains such quadratic tangencies. Combining this with the recent results of Avila-Lyubich-Zhang and former results in collaboration with Lyubich, this yields the abundance of robust homoclinic tangencies in the bifurcation locus for complex Hénon maps. We also study bifurcations induced by families with persistent tangencies, which give another approach to the complex Newhouse phenomenon. (Work in progress)

**Wednesday**

**Victor Kleptsyn:***Hölder regularity of stationary measures*

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

**Hee Oh:***Rigidity of Kleinian groups*

Discrete subgroups of PSL(2,C) are called Kleinian groups. After briefly reviewing the Mostow-Sullivan rigidity theorem, we will discuss new rigidity results for representations of Kleinian groups. (This talk is based on joint work with Dongryul Kim).

**Federico Rodriguez Hertz:***Bounded cohomology and dynamical cocycles*

In this talk I plan to present recent and ongoing work with Pablo Carrasco on the computation of the 2nd bounded cohomology group for hyperbolic groups. The main result is its description in terms of dynamical cocycles with the Bowen property over a "geodesic flow" associated to the group.

**Thursday**

**Thibault Lefeuvre:***Marked length spectrum rigidity for Anosov surfaces*

I will show that on a closed Anosov surface (e.g. a surface with negative sectional curvature), the marked length spectrum, that is the length of closed geodesics marked by the free homotopy of the surface, determines the metric up to isometry. The proof combines hyperbolic dynamics, microlocal analysis and the geometry of complex curves. If time permits, I will also discuss the case of surfaces with boundary. Joint work with A. Erchenko, C. Guillarmou and G. Paternain.

**Karen Butt:***Closed geodesics and stability of negatively curved metrics*

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

**Bryna Kra:***Infinite patterns in large sets of integers*

Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemeredi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used Ergodic Theory to gave a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. We discuss recent developments for infinite patterns leading to the resolution of conjectures of Erdos. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.

**Joel Moreira:***Infinite Sumsets in sets with positive density*

In the 1970's Erdos asked whether every set A of natural numbers with positive upper density has a shift A-t which contains a sumset B+B for an infinite set B of natural numbers? We recently answered this and related questions in joint work with Bryna Kra, Florian Richter and Donald Robertson, using modern tools from ergodic Ramsey theory. I will explain how this result can be formulated as a dynamical question, and outline the main ideas of the proof.

**Mike Hochman:***Strongly irreducible subshifts without periodic points*

A symbolic system is strongly irreducible if there is some g>0 such that any two patterns in the subshift can be glued together as long as they are separated by a gap of size g. This is the strongest mixing condition one can place on a symbolic system, and it implies many good properties, e.g. uniqueness of the measure of maximal entropy, a Krieger-type embedding theorem, and more. In my talk I will discuss the question of the existence of periodic points in such systems, and its connection to a question about periodic points in higher-dimensional shifts of finite type.

**Friday**

**Jialun Li:***On the dimension of limit sets on the real projective plane via stationary measures*

I will present a dimension jump result of limit sets on RP^2 for representations of surface groups in SL(3,R). For Anosov representations, we prove the equality between the Hausdorff dimension and the affinity dimension. In particular, it exhibits a dimension jump under perturbation. The key tool is to study the stationary measures of finitely supported random walks on SL(3,R). We show the Hausdorff dimensions equal the Lyapunov dimensions under certain assumption. This is based on an ongoing joint work with Wenyu Pan and Disheng Xu.

**De-Jun Feng:***Dimension estimates for C^1 iterated function systems*

In this talk, we consider dimensional properties of the attractors of nonlinear and nonconformal iterated function systems (IFS) on R^d. We introduce a generalized transversality condition (GTC) for parameterized families of C^1 IFSs, and show that if the GTC is satisfied, then the Hausdorff and box-counting dimensions of the attractor of a typical IFS are given by the singularity dimension. Moreover, we verify the GTC for some parametrized families of C^1 IFSs. This is based on joint work with Karoly Simon.