Understanding the behavior of the orbits of a general dynamical system is a very hard task. Another approach is to try to understand the average behavior of many orbits. This gives the idea of physical measures, measures that capture the statistical behavior of a set of points having positive volume. An important problem in smooth dynamics is to understand mechanisms that imply the existence of physical measures.
In this talk, we will explore the relation between invariant foliations (stable/unstable foliations), certain types of invariant measures (u-Gibbs/SRB), and how this is connected with the problem of finding mechanisms for the existence of physical measures.

It is an easy observation that the translation flow on an hyperelliptic translation surface is isomorphic to its inverse. We show that this is not the case, when we consider non-hyperelliptic strata. More precisely, we show that there exists a generic subset of every non-hyperelliptic stratum, such that on a translation surface belonging to this subset, the vertical flow is disjoint in the sense of Furstenberg with its inverse.
This is joint work with Krzysztof Frączek and Thierry de la Rue.

For about 2 decades the horocycle flow on strata of translation surfaces was studied, very successfully, in analogy with unipotent flows on homogeneous spaces, which by work of Ratner, Margulis, Dani and many others, have striking rigidity properties. In the past decade Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi proved some analogous rigidity results for SL(2,R) and the full upper triangular subgroup on strata of translation surfaces.

This talk will begin by introducing translation surfaces and describing some of the previously mentioned rigidity before moving onto its goal, that many such rigidity results fail for the horocycle flow on strata of translation surfaces. Time permitting we will also describe some rigidity result for special sub-objects in strata of translations surfaces. This will include joint work with Osama Khalil, John Smillie, Barak Weiss and Florent Ygouf.

Webb showed that the arc complex of a surface of high enough complexity does not satisfy a combinatorial isoperimetric inequality: that is, for every N, there is a loop of length 4 in the arc complex that only bounds discs containing at least N triangles. He showed that the same is true for the free splitting complex and the cyclic splitting complex associated with Out(Fn) and concludes that these complexes do not admit a CAT(0) metric with finitely many shapes. On the contrary, he proved that the curve complex satisfies a linear combinatorial isoperimetric inequality. In this talk, we will show that the free factor complex associated with Out(Fn) also fails to satisfy a combinatorial isoperimetric inequality.

The theory of convex cocompact subgroups of the mapping class group contains two intertwining threads. One is the rich analogy between these subgroups of the mapping class group and the convex cocompact Kleinian group that inspired them. The other is work of Farb and Mosher plus Hamenstädt that shows convex cocompactness is precisely the property that characterizes when an extension of a surface group is Gromov hyperbolic. Both of these threads have natural generalizations that are unresolved. Among Kleinian groups, convex cocompactness is a special case of geometric finiteness, yet no robust notion of geometric finiteness has emerged for the mapping class group. In geometric group theory, there are a variety of generalizations of Gromov hyperbolicity, but there is no characterization of these geometries for surface group extensions. Mosher suggested these two threads should continue to intertwine with geometric finiteness in the mapping class group (however it is eventually defined) being equivalent to some generalization of Gromov hyperbolicity of the corresponding surface group extension. Inspired by their work on Veech groups, Dowdall, Durham, Leininger, and Sisto conjectured that this generalized hyperbolicity could be the hierarchical hyperbolicity of Behrstock, Hagen, and Sisto. We provide evidence for this conjecture by showing that several classes of subgroups that should be considered geometrically finite (stabilizers of multicurves, twist subgroups, cyclic subgroups) correspond to surface group extensions that are hierarchically hyperbolic.

The work of Mann and Rafi gives a classification surfaces Σ when Map(Σ) is globally CB, locally CB, and CB generated under the technical assumption of tameness. We restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann–Rafi, we prove that PMap(Σ) is globally CB if and only if Σ is the Loch Ness monster surface, and locally CB or CB generated if and only if Σ has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures.

The Culler--Vogtmann's Outer space CVn is a space of marked metric graphs, and it compactifies to a set of Fn-trees. Each Fn-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological Fn-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen--Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g-4. In this talk, we give the maximal simplex of length measures on an arational Fn-tree has dimension in the interval [2n-7, 2n-2]. This is a joint work with Mladen Bestvina, Jon Chaika, and Elizabeth Field.

We will discuss about smooth actions on manifolds by higher rank lattices, such as lattices in SL(n,R) with n at least 3. The higher rank property of the acting group suggests that the actions are rigid, which means that the action should have an algebraic origin, such as the Zimmer program and the Katok-Spatzier conjecture. One of the main topics is about how we can provide an algebraic structure on the acting space which is a smooth manifold. We survey some of recent breakthroughs and then we focus on actions on manifolds with “positive entropy” by lattices in SL(n,R), with n at least 3. When the manifold has dimension n, then we will see that the lattice is commensurable to SL(n,Z) from certain "algebraic structure" on M coming from the dynamics. It also gives certain corollaries about the topological properties of M. Part of the talk is ongoing work with Aaron Brown.

A multiarc and curve graph is a simplicial graph whose vertices are arc and curve systems on a compact, connected, orientable surface S. We show that all multiarc and curve graphs preserved by the natural action of PMod(S) and whose adjacent vertices have bounded geometric intersection number are hierarchically hyperbolic spaces with respect to witness subsurface projection. In addition, we prove that the PMod(S)-equivariant quasi-isometry type of such a graph is fully classified by the set of its witness subsurfaces.

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