Realization of cascades on surfaces with finitely many moduli of topological conjugacy
type: Item
instance of: scholarly article
title: Realization of cascades on surfaces with finitely many moduli of topological conjugacy
zbMATH Open document ID: 1287.37013
DOI: 10.1134/S0001434613050271
author: T. M. Mitryakova
author: O. V. Pochinka
published in: Mathematical Notes
publication date: 2013-08-16T00:00:00Z
review text: The paper proves the existence of a surface diffeomorphism, belonging to a particular class, that realizes prescribed dynamical conditions. For a surface \(M^2\), this particular class \(\Psi\) of orientation-preserving diffeomorphisms \(f\in \text{Diff}^r(M^2)\), \(r \geq 5\), consists of those satisfying the following conditions: the nonwandering set \(\Omega_f\) consists of finitely many hyperbolic fixed points; the eigenvalues \(\lambda_p\) and \(\mu_p\) of any saddle point \(p\in \Omega_f\) are positive and satisfy \(\lambda_p^n \mu_p^m\neq 1\) for all \(n,m\in\{ 1,2\}\); if \((W^s_p\setminus p)\cap (W^u_q\setminus q) \neq \emptyset\) for saddle points \(p,q\in \Omega_f\), then \(p\neq q\), and, for any saddle \(r\in \Omega_f\) (including \(p\) and \(q\)), there holds \((W^s_r\setminus r)\cap (W^u_p\setminus p) = \emptyset\) and \((W^s_q\setminus q)\cap (W^u_r\setminus r)=\emptyset\); the wandering set of \(f\) contains finitely many orbits of heteroclinic tangency; and there do not exist saddle points \(p,q\in \Omega_f\) such that all four connected components of the sets \(W^s_p\setminus p\) and \(W^u_q\setminus q\) contain points of one-sided heteroclinic tangency of \(W^s_p\) and \(W^u_q\). The authors of the paper, building on their previous work, develop the notion of the scheme of a diffeomorphism \(f\) in the class \(\Psi\) which is a key to the topological classification of \(f\). The scheme of \(f\) is a set of geometric objects and analytics objects. The geometric objects are related to the topology of the intersection of invariant manifolds of the saddle points. The analytic objects are related to the orbits of one-sided heteroclinic tangency and are given by finitely many moduli of topological conjugacy. The main theorem of the paper is that for every abstract scheme \(S\), there is a surface \(M^2\) of some genus \(g\), and a diffeomorphism \(f_S \in \Psi\) on \(M^2\) whose scheme is equivalent to \(S\).
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37D05
zbMATH DE Number: 6198931
zbMATH Keywords: structurally unstable dynamical system
zbMATH Keywords: cascade
zbMATH Keywords: orbit of heteroclinic tangency
zbMATH Keywords: modulus of topologically conjugacy, scheme of a diffeomorphism
reviewed by: Lennard Bakker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6198931