The reconstruction theorem for endomorphisms
type: Item
instance of: scholarly article
title: The reconstruction theorem for endomorphisms
zbMATH Open document ID: 1032.37012
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2004-03-17T00:00:00Z
review text: 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.
reviewed by: Peter Raith
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 1921216
zbMATH Keywords: endomorphism
zbMATH Keywords: reconstruction
zbMATH Keywords: read out function
zbMATH Keywords: generic
zbMATH Keywords: dimension
zbMATH Keywords: entropy
MaRDI profile type: Publication
full work available at URL: s005740200012
OpenAlex ID: W2049111826
DOI: 10.1007/S005740200012
author: Floris Takens
description: scientific article; zbMATH DE number 1921216