Abstract: 

In the study of circle dynamics, the regularity of a map famously dictates its dynamical behavior—a theme dating back to Denjoy's celebrated C^1 vs. C^2 dichotomy. When considering monotone one-parameter families of circle diffeomorphisms, a natural question arises: how does the regularity of the family govern the regularity of the rotation number as a function of the family parameter? It is known that C^2 families yield a Hölder continuous rotation number with exponent at least 1/2 (Graczyk) and C^1 families guarantee at least log-Hölder continuity (Gorodetski & Kleptsyn). But is this the best we can do for generic C^2 families? And for a generic C^1 family, could Hölder continuity somehow survive? What happens with C^{1+\alpha} families? In this talk, we give partial answers to these questions by establishing a modern analogue to Denjoy's dichotomy: we show that for C2 families, the 1/2-Hölder continuity for the rotation number established by Graczyk is generically optimal, whereas for C1 families, Hölder continuity of the rotation number is generically lost entirely. We also comment on the behavior of families coming from a certain dense subset of the C^{1+\alpha} families. 

Abstract: This is a general introductory talk to the theory of sums and (stable) intersections of Cantor sets: the notion of thickness, Newhouse gap lemma, and a subset of Hall's ray in Lagrange/Markov spectra as an application to the number theory.

In classical (i.e., IID) random matrix dynamics, a question that arises frequently is whether the Lyapunov spectrum is "simple." There are several criteria that imply the existence of a maximal number of distinct Lyapunov exponents for random matrix products and these have been used in various applications. Recently, there have been papers extending this classical theory to specific classes of non-stationary matrix products. In this talk, we will discuss two recent papers ([arXiv:2312.03181] and [arXiv.2507.04058]) which establish gaps between non-stationary analogs of Lyapunov exponents. Strategies and key ideas will be presented, with a brief discussion about applications to Anderson localization.

Abstract: Measure-theoretical and topological entropy play a central role in structural questions for dynamical systems and serve as crucial tools in detecting chaoticity of a system. However, entropy is positive if and only if the system has exponential growth of distinguishable orbit types and it does not provide any information for systems with slower orbit growth. To measure the complexity of systems with subexponential orbit growth several different invariants have been studied. For instance, Anatole Katok and Jean-Paul Thouvenot introduced the concept of slow entropy. In this talk we discuss flexibility results on the values of measure-theoretical slow entropy for rigid transformations and finite-rank systems.

Abstract:

Motivated by the spectral analysis of Sturmian Hamiltonians, we investigate their underlying dynamical systems. This class of subshifts admits a natural extension to a larger family parameterized by the unit interval. In this talk, we study continuity properties of this parametrization with respect to the Hausdorff distance. Starting from the well known fact that these discontinuities arise at rational parameters only, we will pass to a non-Euclidean metric in order to characterize limits at rational parameters. If time permits we briefly discuss immediate implications for Sturmian Hamiltonians.

This talk is based on a joint work with Siegfried Beckus and Jean Bellissard.