Growth of the number of images of a point under iterates of a multivalued map
type: Item
instance of: scholarly article
title: Growth of the number of images of a point under iterates of a multivalued map
zbMATH Open document ID: 0729.58010
DOI: 10.1007/BF01137542
published in: Mathematical Notes
publication date: 1991-01-01T00:00:00Z
review text: The author studies a function \(k\mapsto N_ x(k)=\{y\in X:\) there is a sequence \((x_ 0,x_ 1,...,x_ k)\), \(x_ 0=x\), \(x_ k=y\); \((x_{i- 1},x_ i)\in \Gamma_{\phi}\), \(1\leq i\leq k\}\) where \(\Gamma_{\phi}\) is the graph of a multivalued map (correspondence) \(\phi\) from a set X into itself. The author defines the notions of a symplectic, Liouville integrable, algebraic and abelian correspondence and gives several results concerning the estimation of the number \(N_ x(k)\) for generic points x. The main result states that for an integrable m-valued correspondence there is a constant A such that \(N_ x(k)=Ak^{m-1}\) contrary to the natural expectation that \(N_ x(k)=m^ k\). The paper contains some examples and conjectures. No proofs are given.
Mathematics Subject Classification ID: 58C06
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 4205427
zbMATH Keywords: set-valued mapping
zbMATH Keywords: symplectic manifolds
zbMATH Keywords: Liouville integrability
zbMATH Keywords: correspondence
author: A. P. Veselov
reviewed by: Wojciech Kryszewski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4205427