October 17, 2024 (Thursday), 4:10 p.m.

Darren Creutz, Vanderbilt University

Mike Neamtu, Vanderbilt University

Darren Creutz, Symbolic Dynamics: Connecting short-term and long-term complexity in dynamical systems

Given a dynamical system and a (reasonable) partition of the space, there isa natural map from the system to a symbolic system: assign each element of the partition a distinct ‘letter’, look at the orbit of a given point and ‘read off the infinite word’ by writing the letter of the element of the partition at each time.  Symbolic dynamics is the study of subshifts — closed, shift-invariant subspaces of \mathcal{A}^{\mathbb{Z}} where\mathcal{A} is the ‘alphabet’ — exactly what one obtains from a dynamical system and a partition.

To study the complexity of a subshift (and by extension the complexity of the underlying system it arose from), there are both quantitative and qualitative notions.  Quantitatively, there is the complexity function p(q) = the number of distinct words of length q appearing in any of the infinite words in the subshift.  Qualitatively, there are various notions of mixing and asymptotic independence.

I will present my work (some joint with R. Pavlov and S. Rodock) on word complexity cutoffs for various qualititative mixing properties and (briefly) explore some of the consequences for ‘low complexity’ systems.

 

Mike Neamtu, Convergence of Matrix Products, Subdivision, Refinement Equations, and Cascade Networks

In this talk, we will discuss the following question: Given two square matrices of the same dimension, we consider their arbitrary repeated products and ask under which conditions these products converge to a continuous matrix function. This topic arises from various areas of approximation theory, including subdivision methods for generating curves and surfaces in computer-aided geometric design, refinement equations in multi-resolution analysis, and cascade algorithms. The talk is based in part on joint work with my former student, Diana Sordillo.