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Texas A&M University
Mathematics

Groups and Dynamics Seminar

Spring 2023

 

Date:January 25, 2023
Time:3:00pm
Location:BLOC 302
Speaker:Nataliya Goncharuk, Texas A&M University
Title:Renormalization operators and Arnold tongues
Abstract:Many studies in circle dynamics are devoted to rotation numbers of circle maps, and to so-called Arnold tongues: level sets of rotation numbers in parametric families of circle maps. In particular, E.Risler proved in 1999 that irrational (Diophantine) Arnold tongues in analytic families of circle diffeomorphisms are analytic. In contrast to this result, Llave and Luque observed in 2011 using numerical investigations that Arnold tongues are only finitely smooth near critical circle maps. With M.Yampolsky, we provided explanations of these effects in terms of renormalization operators.

Date:February 1, 2023
Time:3:00pm
Location:zoom
Speaker:Volodymyr Nekrashevych, Texas A&M University
Title:Conformal dimension and other numerical invariants of self-similar groups
Abstract:We will discuss different numerical invariants associated with a self-similar group: l^p-contraction, Ahlfors-regular conformal dimension of the limit space, complexity of portraits, critical exponents for p-parabolicty, for recurrence of random walks, etc. We will show applications of these notions to algebraic properties of the groups and to geometry of the limit spaces.

Date:February 15, 2023
Time:3:00pm
Location:BLOC 302
Speaker:Yury Kudryashov, Texas A&M University
Title:4-periodic orbits in planar billiards
Abstract:Let Ω be a domain in a Euclidean space of dimension d≥2 with sufficiently smooth boundary. Consider the corresponding billiard: a particle moves along straight lines inside Ω and reflects against the boundary of Ω by the standard reflection law. In 1978, Victor Ivrii conjectured that almost every trajectory of this billiard is non-periodic. This condition appeared as a natural geometric assumption in his theorem about asymptotic behavior of the spectrum of the Dirichlet problem for the Laplace operator in Ω. When this conjecture was formulated, people from Sinai's seminar in Moscow predicted it to be solve in a couple of weeks. More than 40 years later, the conjecture still stands.

Date:March 1, 2023
Time:3:00pm
Location:BLOC 302
Speaker:Frank Lin, Texas A&M University
Title:Entropy and quantitative orbit equivalence
Abstract:In the setting of probability measure preserving group actions, two systems are orbit equivalent if their orbits can be matched almost everywhere in a measurable fashion. Orbit equivalence is much weaker than measure conjugacy in general, but various authors have shown that entropy is invariant under certain stronger notions of quantitative orbit equivalence. We introduce some of these results, including (in joint work with Lewis Bowen) the invariance of the f-invariant- a generalization of Kolmogorov-Sinai entropy to actions of free groups - under bounded orbit equivalence.

Date:March 8, 2023
Time:3:00pm
Location:BLOC 302
Speaker:Yaroslav Vorobets
Title:Maximal subgroups of ample (topological full) groups
Abstract:Given a group of homeomorphisms of the Cantor set, usually there are many ways to construct a new homeomorphism, as a puzzle, from pieces of several given ones. The group is called ample (or topological full) if it already contains all homeomorphisms obtained that way. The talk is concerned with the first attempt at a classification of maximal subgroups of ample groups. Results that will be presented are mostly parallel to the classification of maximal subgroups of finite symmetric groups. Recall that all subgroups of the symmetric group are divided into three classes: intransitive subgroups (those that leave invariant a nontrivial subset), imprimitive subgroups (transitive subgroups that leave invariant a nontrivial partition), and primitive subgroups (the remaining ones). It turns out that the maximal intransitive subgroups are stabilizers of certain subsets while the maximal imprimitive subgroups are stabilizers of certain partitions. In the case of the ample groups, arbitrary subsets and partitions are replaced by closed subsets and partitions into closed subsets. Transitivity is replaced by minimality (absence of nontrivial closed invariant subsets). This is the joint work with Rostislav Grigorchuk.