Salvador Addas Zanata

First I will speak about a joint work with Fabio Tal on the dynamics of annulus diffeomorphisms when the annulus is a so-called Birkhoff region of instability. This means that arbitrarily close to each boundary component of the annulus, there are points with positive iterates arbitrarily close to the other boundary component. The main question here is: Do there exist points z and w, such that the alpha limit set of z belongs to one boundary component of the annulus and its omega limit set belongs to the other boundary component, the opposite happening for w? When such points exist, we say that the annulus is a Mather region of instability. In this generality, the answer to the above question is no, that is, there are Birkhoff regions of instability which are not Mather's, even when the rotation set is not reduced to a point (this is a necessary condition for our work). Then, I will apply some of these ideas to the study of generalized Birkhoff regions of instability for C^r generic surface diffeomorphisms (for any r>=1).

Marcelo Alves

The C0-distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms, and a generalisation of the C0-distance on the space of Riemannian metrics. I will explain the following recent result, obtained in collaboration with Lucas Dahinden, Matthias Meiwes and Abror Pirnapasov: the topological entropy of Reeb flows on contact 3-manifolds is lower semicontinuous with respect to the C0 metric on a C-infinity open dense set on the space of Reeb flows. Applied to geodesic flows of Riemannian metrics on surfaces, this says that for "most" Riemannian metrics on closed surfaces, one cannot destroy positivity of topological entropy by C0-small perturbations of the metric. This is in some sense unexpected, as the geodesic flow depends on the derivatives of the Riemannian metric.

Etienne André

Kathryn Mann and Christian Rosendal showed that we can define a quasi-isometry type of homeomorphism groups. The guiding question for the talk will be the following : Can we identify a surface by looking at the quasi-isometry type of its group of homeomorphisms? The following result highlights the difficulty of this question: the homeomorphism group of every surface, apart from the sphere, contains a quasi-isometric copy of the free group.

Pierre Berger

We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we show that these symplectomorphisms can be chosen ergodic or to the contrary with local emergence of maximal order. In particular, this disproves a conjecture of Birkhoff (1941) and solve a problem of Herman (1998). One aspect of the proof provides a new approximation theorem, it enables in particular to implement the Anosov-Katok scheme in new analytic settings.

Christian Bonatti

Following ideas of Mather I recently gave a very large setting where a family of transverse (singular) foliations (or even laminations) of the plane induce a unique compactification of the plane by adding a circle at infinity. Then the end points of these foliations induce "prelaminations of the circle" . In a work in progress with Thomas Barthelmé and Kathryn Mann we characterize the transverse pairs of prelaminations of the circle which are coming for transverse pairs of singular foliations. In this talk I will recall the construction of the circle at infinity and I will present our characterization, together with our motivation for study of Anosov or pseudo-Anosov flows on 3-manifolds.

Jernej Činč

In 2020 Boyland, de Carvalho and Hall published a seminal paper in which they provided a detailed analysis of boundary dynamics of a parameterized family of sphere homeomorphisms with attractors homeomorphic to the (core) tent inverse limit spaces through the study of prime ends. This result presented the first such study of a chaotic parameterized family on a surface. Prior to this result much attention in Continuum Theory has been directed towards the topological classification of tent inverse limit spaces with the highlight being the result by Barge-Bruin-Stimac that for any two different parameters in the parameter range (sqrt{2},2] the tent inverse limits (not restricted to its dynamical core) are non-homeomorphic. Motivated by these result we constructed two parameterized families of sphere homeomorphisms varying continuously with the parameter in [0,1/2] with attractors all homeomorphic to the pseudo-arc and the pseudo-circle respectively, yet presenting rich boundary dynamics. In this talk I will present these constructions that rely on a very useful technique called BBM (Brown-Barge-Martin), which incorporates inverse limits and natural extensions of the underlying bonding maps to embed attractors in manifolds and was presented by Boyland, de Carvalho and Hall. This talk is based on joint works with Piotr Oprocha (AGH Krakow and University of Ostrava).

Odylo Costa

Given a flow on a closed manifold such that all its orbits are periodic, can one prove that the period is uniformly bounded? This was conjectured true until the 70s when Sullivan, Thurston, and Epstein-Vogt gave three different counterexamples to it. In this short talk, we will explain briefly some of those examples.

Andre De Carvalho

Alejo García

Given a torus homeomorphism whose rotation set has nonempty interior, we know it has topological horseshoes, and in particular it has positive topological entropy. If such a map is also C^2, then a recent result by De Carvalho, Koropecki and Tal shows that there exists a uniform bound for the diameter of periodic disks. This yields the existence of a factor with the same rotation set which is obtained by a monotone semiconjugacy, is transitive, area preserving and has dense topological horseshoes. We will see how to adapt this proof to the C^0 context, using the Brouwer-Le Calvez dynamically transverse foliation. This is a joint work with Fábio Armando Tal.

Nancy Guelman

An Interval Exchange Transformation (IET) is a bijection of the unit interval that is piecewise an isometry that preserves orientation. I will talk about the group of IET. In particular we will see some groups that act faithfully by IET and also an obstruction for that kind of r action. In relation to this obstruction I will show different actions of Baumslag Solitar group in low dimension manifolds.

Vincent Humilière

Unlike other similar groups, these groups remained very poorly understood for a long time. I'll review some recent progress in two directions: (non)-simplicity and existence of quasimorphisms. I'll also give some ideas on the proofs which are based on tools from symplectic topology. This is based on joint works with Dan Cristofaro-Gardiner, Cheuk-Yu Mak, Sobhan Seyfaddini and Ivan Smith.

Alejandro Kocsard

Every orientation preserving 2-torus homeomorphism with no periodic orbit is either homotopic to the identity or (conjugate to a homeomorphism homotopic) to a Dehn twist. It is well known that the identity homotopy class is very flexible and contains a very rich variety of different dynamics among periodic point free and minimal homeomorphisms. On the other hand, in this talk we will see that Dehn twist homotopy classes are topologically much more rigid and we shall prove that every minimal homeomorphism in such a homotopy class is a semi-conjugate to an irrational circle rotation. This is a joint work with Ulisses Lakatos (UFF).

Ulisses Lakatos

The action of transitive groups of homeomorphisms encode the symmetries with no privileged reference frames of a manifold. In the case of the unit sphere S², the rotations group SO(3) forms the only such group which is also compact. However, the hierarchy of intermediate closed subgroups between SO(3) and the full group of homeomorphisms is still not completely understood. In particular, it is not known which conditions are sufficient to ensure that such a group contains a map of positive topological entropy. We review a recent result, joint with F. Tal, according to which any proper C¹ extension of PSL(2,C) contains a chaotic map. Time allowing, we also hint at results of a similar nature which may be pursued in the near future.

Pablo Lessa

I will discuss ongoing joint work with Alejo García and Pierre-Antoine Guihéneuf. For any identity isotopy of a closed hyperbolic surface with zero entropy we show that there exists a geodesic lamination which "tracks" the rotation around the surface of generic points with respect to ergodic measures.

Sofía Llavayol

A Quotient of a torus endomorphism (QOTE) f is a self map of the sphere satisfying the following property: there exists a covering map F from the torus to itself and a branched covering map π from the torus onto the sphere such that (f o π)=(π o F). We also ask the degree of F to be grater than one. Every such a map is a Thurston map, meaning a postcritically finite branched covering map of the sphere with degree grater than one. Classical examples of QOTEs are Lattès maps. Associated to a Thurston map there is an orbifold over the sphere, defined from its branching information. I will talk about a joint work with Juliana Xavier, in which we prove that every QOTE has a parabolic orbifold, answering the question of Mario Bonk and Daniel Meyer posed in their book "Expanding Thurston maps".

Kathryn Mann

The fine curve graph of Bowden, Hensel and Webb is an important new tool to understand groups of surface homeomorphisms. I will introduce you to this object and describe some of the relationship between the dynamics of groups acting on the surface, and the dynamics of the induced action on the fine curve graph.

Alejandro Passeggi

Annular dynamics has birth together with the theory of chaos: in the Hamiltonian setting by means of the work of Poincaré on the three body problem and in the dissipative setting by the study of the Van der Pol equation arising in problems involving electric circuits. Since then, several important examples coming from different fields have been reduced to discrete annular dynamics, and the usual question is about whether these systems are integrable or chaotic. Although the mathematical understanding of topological chaos is nowadays quite strong being supported in some renowned models, an important problem still finds a weak answer: given a particular system, decide whether it is chaotic or not. Those systems arising in applications which find a mathematical proof for the existence of chaos are incredibly few, and usually one needs to restrict the parameters so that the system is close to an "integrable" case. In contrast, there is an uncountable list of numerical simulations of systems which are taken as evidence of chaos. In this talk we will comment about the evolution of this problem in view of topological theory of surface dynamics and introduce a result which intends to bring this theory to applicable versions in order to determine the existence of chaos for prescribed systems. In particular, the result creates a bridge that can turn numerical simulations of dynamical systems into rigorous tests for this aim.

José Andrés Rodríguez Migueles

In this talk I state and give examples of the Nielsen-Thurston classification of the isotopy classes of homeomorphism of a surface, via an action on the space of geodesic laminations. Also state Dehn–Nielsen–Baer theorem, which is a rigidity result between topology and algebra in the mapping class group, via an action on the universal cover. And characterize the topological type of some complements of periodic orbits of the geodesic flow via an action on the unit tangent bundle of the surface.

Nelson Schuback

In 1999, Handel classified (up to conjugation) all isotopy classes of Brouwer homeomorphisms (orientation-preserving homeomorphism of the plane without fixed points) relative to one and to two orbits. In other words, he proved the existence of homeomorphisms conjugating Brouwer homeomorphisms so that they belong to certain relative isotopy classes. In 2006, Le Calvez proved that every Brouwer homeomorphism admits a transverse foliation. Such foliations, also defines natural conjugacy maps that allows us to better represent and comprehend the dynamics of the Brouwer homeomorphism. In this work we investigate the links between this conjugacy maps. And as a consequence, we introduce an index between paths transverse to a foliation which recovers Le Roux’s index between orbits of a Brouwer homeomorphism.

Sonja Stimac

I will talk about recent results on topological dynamics of Hénon maps obtained in joint work with Jan Boronski. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The pruning front conjecture (due to Cvitanovic, Gunaratne, and Procacci); A kneading theory (realizing a conjecture by Benedicks and Carleson); A classification: two Hénon maps are conjugate on their strange attractors if and only if their sets of kneading sequences coincide, if and only if their folding patterns coincide. The classification result relies on the further development of the authors’ recent inverse limit description of Hénon attractors in terms of densely branching trees. (Joint work with Jan P. Boronski).

Jing Tao

A cornerstone in low-dimensional topology is the Nielsen-Thurston Classification Theorem, which provides a blueprint for understanding homeomorphisms of compact surfaces. However, extending this theory to non-compact surfaces of infinite type remains an elusive goal. The complexity arises from the behavior of curves on surfaces with infinite type, which can become increasingly intricate with each iteration of a homeomorphism. To address some of the challenges, we introduce the notion of tame maps, a class of homeomorphisms that exhibit non-mixing dynamics. In this talk, I will present some recent progress on extending the classification theory to such maps. This is joint work with Mladen Bestvina and Federica Fanoni.

Pollyana Vicente Nunes

This study is about the relationship between transitivity and topological chaos for homeomorphisms of the two torus. The main result shows that if a transitive homeomorphism of T^2 is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. Moreover, if a transitive homeomorphims of T^2 is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.

Jonguk Yang

The dynamics of real quadratic polynomials is very well understood thanks to the renormalization theory of unimodal interval maps by Sullivan, McMullen, Lyubich and Avila. In this talk, I will present a 2D generalization of this theory which can be applied to the study of dissipative real Hénon maps. In the first part of the talk, I will define a notion of “2D unicriticality,” and then give a survey of the main results (including renormalization convergence). In the second part of the talk, I will describe a key construction used in the proofs of these results called a unicritical cover. This endows the local stable lamination of the renormalization limit set with a 1D structure: a total order that is invariant under the dynamics (except at a unique turning point).