It is a well-known question popularized by Katznelson whether every set of recurrence for all minimal rotations is a set of recurrence for all minimal topological dynamical systems. Positive answers to the question are known for nilsystems and certain skew product systems. On the other hand, in 2022, Glasscock, Koutsogiannis, and Richter showed there exists a skew product where the frequencies in the maximal equicontinuous factor, for some choices of a one dimensional maximal equicontinuous factor, are not responsible for controlling recurrence in the whole system, revealing the presence of hidden frequencies. In this talk, we show the same phenomenon happens for all choices of a finite dimensional maximal equicontinuous factor and more generally for a broad class of skew products.
We investigate quantitative recurrence in dynamical systems, focusing on the asymptotic behavior of return-time statistics to shrinking target sets. Our goal is to describe how the distribution of return times evolves as the targets become small and to identify the corresponding limiting processes. In uniformly and a broad class of non-uniformly hyperbolic systems preserving a finite measure, the limiting process is typically Poissonian. This gives rise to a well-known dichotomy: non-periodic points yield a standard Poisson process, whereas periodic points lead to a compound Poisson process. We extend this analysis to infinite measure-preserving systems. In this setting, the classical Poisson limit is replaced by the fractional Poisson process, reflecting the underlying null-recurrent dynamics. Moreover, in the context of null-recurrent countable Markov shifts, we identify the limiting processes for all points, revealing a trichotomy that replaces the classical periodic/non-periodic dichotomy observed in the finite-measure case by adding a new family of points leading to other limits.
In 2019 Moreira, Richter, and Robertson used ergodic theory to resolve a conjecture of Erdős, proving that every set of natural numbers with positive upper Banach density contains a subset of the form B+C for some infinite sets B and C. Shortly after, Host showed that one cannot insist that B or C have positive density. In their survey paper, Kra, Moreira, Richter, and Robertson ask if any type of growth rate can be required of the sets B and C. Additionally they ask questions about asymptotics for the growth of finite sumsets. I will discuss joint work with Felipe Hernández regarding some answers to the questions of Kra, Moreira, Richter, and Robertson.
Motivated by Bourgain's return times theorem, recently, Donoso, Maass, and Arya-Saavedra studied ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems. In this talk, we will show that their results can be improved to multiple ergodic averages for commuting transformations, and a wider range of sequences. This talk is based on joint work with Sebastián Donoso, and Vicente Saavedra-Araya.
Anderson localization is a physical phenomenon that was observed by Phillip Anderson. One definition of localization is when the spectrum of Schrodinger operator has pure point spectrum with exponentially decaying eigenfunctions. The Lyapunov exponent plays a central role in studying the phenomenon, as uniformly positive Lyapunov exponents paired with a large deviation estimate has been a large indication of localization. Our focus is when one can prove uniform positivity, as Kotani theory would imply that the family of Schrodinger operators has empty absolutely continuous spectrum. In our talk, we briefly discuss the setting and the methods used to show uniform positive Lyapunov exponents for lower Holder potentials along local unstable leaves when generated by hyperbolic dynamics with at least one expanding direction.
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Irreducible higher rank Anosov actions often exhibit strong rigidity properties. Katok and Spatzier conjectured that such actions are always smoothly conjugate to algebraic actions. This conjecture, without additional assumptions, was shown to be false by Vinhage. To construct his counterexample, he began with products of Anosov flows, which despite being reducible at first, can give rise to irreducible actions via nontrivial time changes. In this talk, we present two rigidity results for time changes of products of Anosov flows related to stabilizers of periodic orbits. In our first theorem, stabilizers are used to detect when two time changes are conjugate, while in our second theorem they detect when the resulting action is reducible. This is joint work with Miri Son.
Given a translation surface with uniquely ergodic vertical flow, one can construct a branched slit-induced double cover by taking two copies of the surface, making identical slits on each copy, and gluing them crosswise. We study when the vertical flow on the resulting surface is uniquely ergodic. We prove a geometric criterion for this property and show that almost every choice of the slit produces a uniquely ergodic double cover. This talk is based on joint work with Polina Baron.