The reconstruction theorem for endomorphisms

Abstract

The reconstruction theorem deals with dynamical systems which are given by a map φ : M → M together with a read out function ƒ : M → ℝ. Restricting to the cases where ψ is a diffeomorphism, it states that for generic (φ,ƒ) there is a bijection between elements x ∈ M and corresponding sequences (ƒ(x),ƒ (φ(x)), . . . , ƒ (φ^{k -1}(x))) of k successive observations, at least for k sufficiently big. This statement turns out to be wrong in cases where ψ is an endomorphism.

In the present paper we derive a version of this theorem for endomorphisms (and which is equivalent to the original theorem in the case of diffeomorphisms). It justifies, also for dynamical systems given by endomorphisms, the algorithms for estimating dimensions and entropies of attractors from obervations.