Growth of the number of images of a point under iterates of a multivalued map
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.
scientific article; zbMATH DE number 4205427