- Ergodic Theory and Dynamical Systems Seminar
- CUNY Graduate Center
- Fridays 11:45 AM -12:45 PM, Room 5417 (Learning Seminar) and 3:00 PM - 4:00 PM, Room 6417 (ETDS Seminar)
- Organizers: Yunping Jiang, Tamara Kucherenko, and Enrique Pujals
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Enrique Pujals , CUNY Graduate Center, New York
Title: Dynamical Shadowing and Finite-Dimensional Approximation of Smooth Diffeomorphisms
Abstract: This first part introduces key concepts in the dynamical approximation of smooth diffeomorphisms on compact manifolds. We review notions such as pointwise ε-shadowing and average shadowing, and classical results like shadowing in hyperbolic dynamics. We also examine the geometric structure of the space of bounded Cr-diffeomorphisms (or endomorphisms) and the idea of finite-dimensional submanifolds as theoretical tools to approximate general dynamics. This session provides the necessary background and intuition for the more speculative discussion in Part II.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Enrique Pujals , CUNY Graduate Center, New York
Title: Constructing Finite-Dimensional Manifolds for Dynamical Approximation
Abstract: This second part focuses on the mathematical problem of constructing finite-dimensional manifolds of diffeomorphisms (or endos) that can dynamically approximate any bounded smooth map under r average shadowing. We explore candidate constructions based on hyperbolic blocks, piecewise-affine maps, and other dynamical building blocks, illustrating how these structures could serve as finite-dimensional “cores” capable of reproducing complex orbit behavior. We then discuss how such candidate manifolds could be realized within finite parametric families, for example via neural networks, transformers, or programmatic models, connecting rigorous dynamical constructions to potential computational implementations. Open questions concern the dimension and structure required for these manifolds and the treatment of systems with mixed hyperbolic and neutral dynamics.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Bryce Gollobit , CUNY Graduate Center, New York
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Be'eri Greenfeld , CUNY Hunter College, New York
Title: How complicated can words be?
Abstract: Consider an infinite word over a finite alphabet, or more generally a subshift (a symbolic dynamical system). A quantitative measure of its combinatorial complexity is given by the complexity function, which counts, for each natural number n, the number of distinct subwords of length n that occur in it. This fundamental notion is deeply connected to dynamical systems, computer science, Diophantine analysis, and more.
We solve the "inverse problem" of determining which functions can arise (asymptotically) as complexity functions of infinite words and subshifts. We further investigate the range of possible complexities under additional dynamical assumptions such as minimality and unique ergodicity. As applications, we obtain consequences for the convergence of random graphs, and for the theory of growth in semigroups and algebras.
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Hiroki Takahasi , Keio University, Japan
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Hiroki Takahasi , Keio University, Japan
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
TBA
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Maria Sabitova, CUNY Queens College, New York, NY
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Ronni Pavlov , University of Denver, Denver, CO
Title: TBA
Abstract: TBA
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Ronni Pavlov, University of Denver, Denver, CO
Title: TBA
Abstract: TBA
Learning Seminar 11:45 AM-12:45 PM, GC 5417
Barak Weiss , Tel Aviv University, Israel
Title: A geometrical counting problem arising in billiards, and a dynamical approach
Abstract: I will discuss what is known about counting saddle connection holonomies in translation surfaces, and what is conjectured. I will describe a dynamical approach to such problems, following Eskin, Masur and Veech. This reduces the counting problem to some equidistribution question on the moduli space of translation surfaces.
[1] R. Mãńe, P. Sad and D. P. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217.
Ergodic Theory and Dynamical Systems Seminar, 3:00 PM - 4:00 PM, GC 6417
Qiaochu Ma , Texas A&M University, College Station, TX
Title: Mixed quantization and quantum ergodicity on graph vector bundles
Abstract: Quantum Ergodicity (QE) is a classical topic in quantum chaos. It asserts that on a compact Riemannian manifold whose geodesic flow is ergodic, the Laplacian admits a density-one subsequence of eigenfunctions that become equidistributed. In this talk, we present a discrete version of QE for vector bundles over regular graphs. The main technique is mixed quantization, which brings together semiclassical and geometric quantizations.