Takens-type reconstruction theorems of one-sided dynamical systems

Abstract

The reconstruction theorem deals with dynamical systems that are given by a map of a compact metric space X together with an observable from X to the real line . In 1981, by use of Whitney’s embedding theorem, Takens proved that if is a (two-sided) diffeomorphism on a compact smooth manifold M with , for generic (T, f) there is a bijection between elements and corresponding sequence , and moreover, in 2002 Takens proved a generalised 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 ‘non-invertible’ dynamical systems for a large class of compact metric spaces, which contains PL-manifolds, manifolds with branched structures and some fractal sets, e.g. Menger manifolds, Sierpiński carpet and Sierpiński gasket and dendrites, etc.