Takens-type reconstruction theorems of one-sided dynamical systems on compact metric spaces

TL;DR

This paper extends Takens' reconstruction theorems from smooth manifolds to a broad class of compact metric spaces, including fractals and dendrites, using topological methods.

Contribution

It generalizes Takens' theorems to one-sided dynamical systems on complex spaces like fractals and branched manifolds, beyond smooth manifolds.

Findings

Reconstruction theorems for compact metric spaces including fractals.

Application of topological methods to extend existing theorems.

Inclusion of spaces like Sierpiński carpet and dendrites.

Abstract

The reconstruction theorem deals with dynamical systems that are given by a map $T:X\to X$ of a compact metric space $X$ together with an observable $f:X \to \R$ from $X$ to the real line $\R$. In 1981, by use of Whitney's embedding theorem, Takens proved that if $T:M\to M$ is a diffeomorphism on a compact smooth manifold $M$ with $\dim M=d$, for generic $(T,f)$ there is a bijection between elements $x \in M$ and corresponding sequence $(fT^j(x))_{j=0}^{2d}$, and moreover, in 2002 Takens proved a generalized version for endomorphisms. In natural sciences and physical engineering, there has been an increase in importance of fractal sets and more complicated spaces, and also in mathematics, many topological and dynamical properties and stochastic analysis of such spaces have been studied. In the present paper, by use of some topological methods we extend the Takens' reconstruction theorems of compact smooth manifolds to reconstruction theorems of one-sided dynamical systems for a large class of compact metric spaces, which contains PL-manifolds, branched manifolds and some fractal sets, e.g. Menger manifolds, Sierpi\'nski carpet and Sierpi\'nski gasket and dendrites, etc.