Breadcrumb
Sharp Embedding Theorems for Topological Dynamical Systems
Thu 31 January 2013, 15:00
Yonatan Gutman
University of Cambridge
Ergodic Theory and Dynamical Systems
Organisers: Andrew Ferguson, Thomas Jordan
ABSTRACT
Given a metric space $X$ of dimension $d$ it is a classical
fact that the minimal $n$ that guarantees that $X$ can be embedded in
$[0,1]^{n}$, is $n=2d+1$. An analogous problem in the category of
dynamical systems, is under what conditions one can guarantee that
the metric topological system $(X,T)$ is embeddable in
$([0,1]^{n})^{\mathbb{Z},shift}$. The embedding is induced by $n$ real
continuous functions on $X$, so this question can also be thought as a
topological dynamical analogue of the Krieger Generator Theorem. A
well known theorem by Jaworski states that if $(X,T)$ is aperiodic and
$X$ is finite dimensional then $n=1$ is sufficient. Lindenstrauss and Tsukamoto have suggested a far reaching and sharp generalization of this theorem, involving mean dimension, an invariant distinguishing systems of infinite entropy. I will discuss classes of dynamical systems where their conjecture has been verified.ÂÂ