## Courses

**Speaker**: Vaibhav Gadre

**Title**: Geometry and dynamics of flat surfaces

**Summary**: This mini-course will be an introduction to the geometry and dynamics of flat surfaces, culminating in the most recent advances and some open problems in the field. The course will begin by explaining some of the motivations for the interest in flat surfaces. These include the dynamics of interval exchange maps, rational billiards, windtree model, etc. The course will then proceed to outline the basic theory of flat surfaces and renormalisation techniques that relate problems of dynamics on flat surfaces to the dynamics on their moduli spaces given by the \(\mathrm{SL}(2,\mathbb{R})\) action. We will give a quick summary of the definitive work by Eskin–Mirzakhani that established Margulis–Ratner type results for \(\mathrm{SL}(2,\mathbb{R})\) orbit closures. We will then proceed to the dynamics of the Teichmüller flow given by action of the diagonal part of \(\mathrm{SL}(2,\mathbb{R})\). Time permitting, we will sign off by listing some open problems in the field.

**Speaker**: Bryna Kra

**Title**: Infinite patterns in large sets of integers

**Summary:** Resolving a conjecture of Erdos and Turan from the 1930’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 must occur in sufficiently large sets of integers and an understanding of what types of structures control these behaviors. Only recently have we been able to extend these methods to infinite patterns, resolving conjectures of Erdos from the 1970s. This series of lectures will cover these new developments, leading to numerous open problems as to what other infinite configurations occur in sufficiently large sets of integers. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.

**Speaker**: Scott Schmieding

**Title**: Automorphism groups of symbolic systems

**Summary:** The automorphism group of a topological dynamical system consists of all homeomorphisms of the underlying space which commute the base dynamics. In the context of symbolic systems, these groups form a rich collection of countable groups whose structure can vary dramatically depending on the complexity of the underlying dynamics. The automorphism groups of shifts of finite type in particular have been heavily studied, beginning with Hedlund and others in the late 60’s, yet remain highly mysterious. More recent work has focused on these groups in the context of low complexity systems, where their structure is much more constrained. These lectures will give an introduction to these areas, as well as the recently introduced stabilized automorphism groups, along with some newer results on these in the context of shifts of finite type.

**Speaker**: Wenbo Sun

**Title**: Partition regularity problems for quadratic function

**Summary**: A well-known result of Roth says that every subset of integers with positive density contains non-trivial solutions to the linear equation \(x-2y+z=0\). In this case, we say that the equation \(x-2y+z=0\) is partition regular. Nowadays, partition regularity results for linear equations are well studied. On the contrary, very little progress was made for nonlinear equations. In these lectures, we will discuss some recent progress on the partition regularity results for quadratic equations with three variables.

## Talks

**Speaker**: Francisco Arana-Herrera

**Title**: Effective mapping class group dynamics

**Summary**: Motivated by counting problems for closed geodesics on hyperbolic surfaces, I will present a family of new results describing the dynamics of mapping class groups on Teichmüller spaces and spaces of closed curves of closed surfaces.

**Speaker**: Mauro Artigiani

**Title**: Rigidity for time-changes of unipotent flows.

**Summary**: Parabolic flows form an intermediate category between elliptic and hyperbolic flows. They exhibit some characteristic associated with non-chaotic systems, and some associated with highly chaotic ones. A fundamental example is the horocycle flow on hyperbolic surfaces and, more generally, homogeneous flows generated by multiplication by a unipotent element on a Lie group. It is much more difficult to produce examples of non homogeneous parabolic flows, since perturbations usually lead to hyperbolic flows.

The simplest perturbation, still in the parabolic realm, is given by time-changes. These have been investigated in detail in the case of the horocycle flow and for nilflows. In this talk, we will give a detailed introduction to the classical theory of horocycles and their time-changes, before presenting our result, joint with Livio Flaminio and Davide Ravotti, on rigidity of time-changes of unipotent flows on finite volume quotients of simple Lie groups, which generalizes Ratner’s classical work on the horocycle flow.

**Speaker:** Paulina Cecchi-Bernales

**Title**: Coboundaries and eigenvalues for finitary \(S\)-adic shifts.

**Summary**: An \(S\)-adic shift is a symbolic system obtained by performing an infinite composition of morphisms defined over possibly different finite alphabets. It is said to be *finitary* if these morphisms are taken from a finite set. \(S\)-adic systems are a generalization of substitution shifts. In this talk we will discuss spectral properties of finitary \(S\)-adic shifts. Our departure point will be a theorem by B. Host which characterizes eigenvalues of substitution shifts, and where *coboundaries* appear as a key tool. We will introduce the notion of \(S\)-adic coboundaries and present some results which show how they are related with eigenvalues of \(S\)-adic shifts. This is joint work with Valérie Berthé and Reem Yassawi.

**Speaker**: María Isabel Cortez

**Title** On a family of subshifts that characterizes the residually finite groups

**Summary**: Toeplitz subshifts can be characterized as the almost 1-1 extensions of equicontinuous and minimal actions on the Cantor of residually finite groups. They were introduced in the context of \(\mathbb{Z}\)-actions in 1969 by Jacob and Keane. Since then, they have proved to be a versatile class of dynamical systems in which it is possible to study different dynamical behaviors. In this talk we will discuss some of their properties, in particular, some recent results regarding their space of invariant measures (joint work with Paulina Cecchi-Bernales, Jaime Gómez and Samuel Petite).

**Speaker**: Sebastián Donoso

**Title**: Expansivity in topological dynamics: a geometrical framework via Robinson Crusoe theorem.

**Summary**: For a countable group \(G\) acting continuously on a compact metric space, the Robinson Crusoe theorem asserts there exist two orbits staying at an arbitrary small distance along some horoball of \(G\). Moreover, under some mild conditions, the horoball can be chosen a priori in a restricted half-space. This result extends similar ones of Schwartzman for \({\mathbb Z}\)-action and Boyle-Lind for \({\mathbb Z}^d\) -action, which leads to the notion of nonexpansive direction. We derive several applications in topological and Cantor dynamics that are novel even when the acting group is \({\mathbb Z}^d\). For instance, we show that in any half-space of \({\mathbb R}^d\) there exists a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind. This is a joint work with A. Maass and S. Petite.

**Speaker**: Bastián Espinoza

**Title**: An \(S\)-adic characterization of sublinear complexity subshifts

**Summary**: In the context of symbolic dynamics, the class of “sublinear complexity subshifts” is of particular relevance as it occurs in a variety of areas, such as geometric dynamical systems, language theory, number theory, and numeration systems, among others. During the intensive study carried on this subject from the beginning the 90’s, it was proposed that a hierarchical decomposition based on \(S\)-adic sequences that characterizes sublinear complexity subshifts would be useful to understand this class. The problem of finding such a characterization was given the name “\(S\)-adic conjecture” and inspired several influential results in symbolic dynamics. In this talk, I will present an \(S\)-adic characterization of sublinear complexity subshifts and some of its applications, which in particular give a solution to this conjecture.

**Speaker**: Hélène Eynard-Bontemps

**Title**: Deformation by conjugation of diffeomorphisms of the interval

**Summary**: The study of the pathconnectedness of the space of \(C^r\) actions of \(\mathbb{Z}^2\) on the interval [0,1] plays an important role in the classification of codimension 1 foliations on 3 manifolds. One way to deform actions is by conjugation. If an action can be brought arbitrarily close to the trivial one by conjugation, it is said to be quasi-reducible. In this talk, we will present and compare obstructions to quasi-reducibility in different regularity classes, and draw conclusions concerning the initial connectedness problem (joint with A. Navas).

**Speaker**: Felipe García-Ramos

**Title**: Completeley positive mean dimension

**Summary**: Mean dimension is a dynamical invariant introduced by Gromov particularly useful for studying systems with infinite entropy. A dynamical system has completely positive mean dimension if every non-trivial factor has positive mean dimension. We study the complexity of the set of dynamical systems with completely positive mean dimension.

Joint work with Yonatan Gutman.

**Speaker**: Katrin Gelfert

**Title**: Mingled hyperbolicities: Restricted variational principles.

**Summary**: We aim to quantify the “lack of hyperbolicity” in transitive nonhyperbolic diffeomorphisms. For that we study skew products of \(C^1\) circle diffeomorphism which, in some sense, capture the key mechanism of nonhyperbolic behavior of robustly transitive dynamical systems. They naturally arise, for example, from the projective action of certain \(2 \times 2\) elliptic matrix cocycles. The coexistence of saddles of different types of hyperbolicity is described in terms of fiber-expanding and -contracting regions which are mingled by the dynamics. It gives also rise to nonhyperbolic ergodic measures which are characterized in terms of a zero Lyapunov exponent in the circle fiber-direction. We perform a multifractal analysis for fiber-Lyapunov exponents and establish restricted variational principles of the topological entropy of the level set of each exponent in terms of the metric entropy of ergodic measures. In particular, we describe the maximal entropy of ergodic nonhyperbolic measures. This is joint work with L.J. Díaz and M. Rams.

**Speaker**: Rodolfo Gutiérrez-Romo

**Title**: The flow group of rooted abelian or quadratic differentials

**Summary**: Given a stratum component of rooted abelian or quadratic differentials \(C\), we investigate the connection between its fundamental group and the dynamics of the Teichmüller flow by means of the “flow group”; this group captures the part of the fundamental group that is “detected” by said flow. We show that the flow group equals the fundamental group, which can be interpreted as the dynamics detecting the entirety of the topology of \(C\). Our main application is establishing the conjecture by Kontsevich and Zorich stating that the Lyapunov spectrum of the Kontsevich–Zorich cocycle is simple.

This is joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre and Saul Schleimer.

**Speaker**: Arnaldo Nogueira

**Title**: Rotation number of 2-interval piecewise affine maps.

**Summary**: Let \(f=\displaystyle f_{\lambda,\delta,\mu}\) be a family of interval maps described by the below graph in terms of the parameters \(\lambda,\delta,\mu\). Any map \(f\) has a rotation number and its dynamics will be described with the help of two functions of the three parameters whose definitions use Hecke-Mahler series. As a consequence of our approach, we prove that whenever the parameters \(\lambda,\delta\) e \(\mu\) are algebraic numbers the rotation number takes a rational value. This result extends our previous theorem about the case where \(f\) is a circle contracted rotation which means that the parameter \(\mu=1\). The talk is based on a joint work with Michel Laurent.

**Speaker**: Felipe Riquelme

**Title**: Exceptional sets for geodesic flow on non-compact varieties.

**Summary**: An exceptional set is composed of points in a space whose orbits do not accumulate in a given set. The interest of studying the size of these sets lies in the following phenomenon that is presented to us in various contexts: if the given set is small, its exceptional set must be large. For example, in the case of subshifts of finite type, Dolgopyat proved that sets of entropy less than the entropy of the system have exceptional sets of total entropy. In the same work, Dolgopyat also proves that for Anosov diffeomorphisms of the torus, sets of dimension less than the dimension of the SRB measure have exceptional sets of total dimension. For continuous dynamics we highlight the work of Kleinbock who proved that, for geodesic flow, the exceptional set of a compact proper subvariety of a compact variety at negative curvature has full dimension. In this talk we will study the case of exceptional sets for geodesic flow in noncompact varieties, proving that sets of entropy less than the entropy of the system have exceptional sets of total entropy. This is a collaborative work with Katrin Gelfert (UFRJ, Brazil).

**Speaker**: Frank Trujillo

**Title**: Hausdorff dimension for invariant measures of circle homeomorphisms with breaks

**Summary**: By a classical theorem of Denjoy, any sufficiently regular piece-wise smooth circle homeomorphism with finitely many branches (often called a *circle homeomorphism with breaks*) and irrational rotation number is topologically conjugate to an irrational circle rotation. In particular, it admits a unique invariant probability measure.

We will discuss dimensional properties of this measure and show that, generically, this unique invariant probability measure has zero Hausdorff dimension. To encode this generic condition, we consider piece-wise smooth homeomorphisms as generalized interval exchange transformations of the interval and rely on the notion of *combinatorial rotation number*.

**Speaker**: Ferrán Valdez

**Title**: Topological dynamics in big mapping class groups.

**Summary**: We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface \(S\) on itself. We will explain the ideas and definitions behind the proof of the following results:

(1) For any infinite type surface \(S\), all conjugacy classes of \(\mathrm{MCG}(S)\) are meager.

(2) \(\mathrm{MCG}(\Sigma)\) has a dense conjugacy class if and only if \(\Sigma\) has a unique maximal end and no non-displaceable finite-type subsurfaces.

**Speaker**: Aníbal Velozo

**Title**: The Hausdorff dimension of points that diverge on average for the geodesic flow

**Summary**: A remarkable result of Bishop and Jones states that the Hausdorff dimension of the radial limit set of a Kleinian group is equal to the critical exponent of the group. Moreover, Otal and Peigne proved that the critical exponent of the group is equal to the topological entropy of the geodesic flow on the quotient manifold. These results illustrate a connection between entropy, Hausdorff dimension and critical exponents in this context. In this talk I will discuss analogous results for the entropy at infinity of the geodesic flow, which accounts for the entropy accumulation at the ends of the manifold, and its relation to the Hausdorff dimension of points that diverge on average. This is joint work with Felipe Riquelme.