In my talk I shall present my work with Sonja Štimac on Hénon maps with strange attractors (Wang-Young parameters). First I shall explain a construction (inspired by a work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane) of conjugacy of these maps to the shift homeomorphisms on inverse limits of dendrites with dense set of branch points, and a characterization of orbits of critical points in terms of these inverse limits. Then I will explain how this leads to a classification of conjugacy classes of such maps in terms of a single sequence of 0s and 1s.

References:

1. Boronski J., Štimac S; Densely branching trees as models for Hénon-like and Lozi-like attractors, Advances in Mathematics 429 (2023) 109191
2. Boronski J., Štimac S; The pruning front conjecture, folding patterns and classification of Hénon maps in the presence of strange attractors, arXiv:2302.12568v2

Acylindricity may be viewed as a generalization of being a uniform lattice in a locally compact second countable group. The theory of acylindrical actions on hyperbolic spaces has seen an explosion in recent years. Trees are of course examples of hyperbolic spaces, and by considering products, we start to see new and interesting behaviors that are not present in rank-1, such as the simple Burger-Mozes-Wise lattices, or Bestvina-Brady kernels.

In a joint work with S. Balasubramanya we introduce a new class of nonpositively curved groups. Viewing the theory of S-arithmetic semi-simple lattices as inspiration, we extend the theory of acylindricity to higher rank and consider finite products of $\delta$δ-hyperbolic spaces. The category is closed under products, subgroups, and finite index over-groups. Weakening acylindricity to AU-acylindricity (i.e. acylindricity of Ambiguous Uniformity) the theory captures all $S$S-arithmetic semi-simple lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, and many others. In this talk, we will discuss structure theorems such as the Tits' Alternative. This structure allows us to give a partial resolution to a conjecture by Sela.

Many of us remember when the Spring Topology Conference became the the Spring Topology and Dynamics conference in part because continua theorists were finding so many things they wanted to work on in dynamics. Classical Interval Dynamics is now a mature field with hundreds of articles and many books. Years ago continua theorists with considerable inspiration from Devaney’s accessible book began extending theorems in interval dynamics like the Sarkovski theorem to chainable continua, that is continua that are the inverse limit of interval functions. A favorite tool of continua theorists, inverse limits, have also been used in dynamical systems since whatever one might call the beginning.

Inspired by Ethan Akin our group has been constructing continua and functions at the same time that have a variety of dynamical properties using what we call Mahavier products, also known as an inverse limit with a set valued function. Akin would probably call what we are doing the dynamics of closed relations. The dynamics are those of shift maps. In other words we extend from the now classic topic of dynamics of shift maps on inverse limits with a single bonding map to continua that cannot be expressed as an inverse limit with a single continuous function on a simpler space like an arc or a tree or a circle, but can be expressed as an inverse limit with a single closed relation. Specifically in this talk we look at various ways to express the Cantor fan and the Lelek fan as a Mahavier products . We obtain transitive homeomorphisms, mixing homeomorphisms, with and without a dense set of periodic orbits and with zero or positive entropy. This is joint work with Iztok Banic, Judy Kennnedy, Chris Mouron, and Goran Erceg.

We shall give a brief survey of the already rich theory of Turkey reductions between directed sets. The emphasis will be given to application to set-theoretic topology with cardinal restrictions replaced by Tukey reductions.

The mapping class group is the group of orientation preserving homeomorphisms of a surface up to isotopy. In particular, the mapping class group encodes information about the symmetries of a surface. The Nielsen-Thurston classification states that elements of the mapping class group are of one of three types: periodic, reducible, and pseudo-Anosov. In this talk, we will focus our attention on the pseudo-Anosov elements, which are the elements of the mapping class group which mix the underlying surface in a complicated way. In this talk, we will discuss both classical and new results related to pseudo-Anosov mapping classes, as well as the connections to other areas of mathematics.