  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.
