Abstracts

Anibal Velozo

Title: How much chaos is there at infinity?
Abstract: In this talk the notion of entropy at infinity and its appearance in non-compact contexts will be discussed: symbolic dynamics and the geodesic flow on negatively curved manifolds. Roughly speaking, the entropy at infinity quantifies the dynamical complexity at infinity; analog to how the topological entropy quantifies the entropy of the recurrent part of the system. I will outline the main features of this entropy and how it plays a fundamental role in the ergodic theory of non-compact systems via two of the prime examples in hyperbolic/expanding dynamics. This talk is partially based on joint works with G. Iommi, F. Riquelme, and M. Todd.

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Tsviqa Lakrec

Title: Affine random walks on the torus and on nilmanifolds.
Abstract: I will discuss two quantitative equidistribution results for the random walks on a torus and on a nilmanifold arising from the action of the groups of affine transformations for each space. This is a joint work with Weikun He and Elon Lindenstrauss.

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Snir Ben Ovadia

Title: Hyperbolic SRB measures and the leaf condition.
Abstract: Hyperbolic SRB (after Sinai, Ruelle, and Bowen) measures are an important object in dynamical systems, as a substitute to the Liouville measure for non-closed systems. The question of their existence is still a major open problem. We give a sufficient and necessary condition for their existence in the form of a leaf condition-a geometric condition.

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Nattalie Tamam

Title: Distribution of discrete groups orbits.
Abstract: Distribution of discrete groups orbit is a classical question that is deeply connected to the distribution of horospherical flows in a related homogeneous space. We will give effective equidistribution results for both when the homogeneous space has infinite volume. This is a joint work with Jacqueline Warren.

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Andreu Moragues

Title: Totally Ergodic and Weakly Mixing sets.
Abstract: In his proof of Szemerédi’s theorem, Furstenberg established that given a large subset \(E \subseteq \mathbb{Z}\), we can study its patterns by associating a measure preserving system \((X,\mathcal{B},\mu,T)\) to \(E\). In this talk we will discuss combinatorial conditions on \(E\) that ensure that the system associated to it has good mixing properties (such as total ergodicity and weak mixing) and analyze some of its consequences.

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Mahbub Alam

Title: Intrinsic Diophantine Approximations on Spheres.
Abstract: We will see an application of ergodic theory and dynamics to number theory. Dynamics on quotients of lie groups have found numerous applications in number theory – examples include Oppenheim’s conjecture, Littlewood’s conjecture, Diophantine approximation on Rn. Recently Kleinbock and Merrill have used dynamics to understand Diophantine approxima- tion on spheres. I will talk about how one can model Diophantine approximation on spheres in terms of dynamics and mention a quantitative result on this topic that my supervisor Prof. Anish Ghosh and I have proved. Afterwards I will show how Birkhoff’s ergodic theorem can give us counting results in Diophantine approximation.

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