Realization of cascades on surfaces with finitely many moduli of topological conjugacy
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 \(S\).
modulus of topologically conjugacy, scheme of a diffeomorphism
scientific article; zbMATH DE number 6198931
scientific article