Takens-type reconstruction theorems of one-sided dynamical systems

The reconstruction theorem deals with dynamical systems that are given by a map T : X -> X of a compact metric space X together with an observable f : X-* R from X to the real line R. In 1981, by use of Whitney's embedding theorem, Takens proved that if T : M -> M is a (two-sided) diffeomorphism on a compact smooth manifold M with dimM = d , for generic (T , f) there is a bijection between elements x is an element of M and corresponding sequence (fT(j)(x))(2d)(j=0) , and moreover, in 2002 Takens proved a generalised version for endomorph-isms. In natural sciences and physical engineering, there has been an increase in importance of fractal sets and more complicated spaces, and also in math-ematics, 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, man-ifolds with branched structures and some fractal sets, e.g. Menger manifolds, Sierpinski carpet and Sierpinski gasket and dendrites, etc.