The reconstruction theorem for endomorphisms
Consider a \(C^{1}\)-endomorphism \(\varphi :M\to M\) on a compact \(C^{1}\)-manifold \(M\) together with a \(C^{1}\)-function \(f:M\to{\mathbb R}\) (called the read out function). Suppose that \(k\) is a (fixed) sufficiently large number. The problem addressed in this paper is if there is a one-one correspondence between \(x\in M\) and the vector \((f(x),f(\varphi x),\dots ,f(\varphi^{k-1}x))\). This means is it possible to ``reconstruct'' \(x\), if we can only measure the values of the read out function during the first \(k\) times. In the case of diffeomorphisms it is known that for a generic \(C^{1}\)-diffeomorphism \(\varphi\) the answer to the problem posed above is positive. This result is called the reconstruction theorem. On the other hand, it is known not to hold for \(C^{1}\)-endomorphisms in general. Under some technical, but reasonable conditions the author is able to prove the reconstruction theorem also for endomorphisms. As all diffeomorphisms satisfy these conditions this result is a generalization of the reconstruction theorem for diffeomorphisms. Some standard techniques for calculating entropy and dimension make use of the reconstruction theorem. Hence these techniques can also be applied for endomorphisms satisfying the assumptions of the reconstruction theorem.
scientific article; zbMATH DE number 1921216