Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle

Abstract

This paper introduces class G containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of 2q periodic points, where q is some natural number. So Cartesian products of two such diffeomorphisms has 4q1q2 periodic points where q1 corresponds to the first transformation and q2 corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains 2q1q2 saddle points, q1q2 sinks, and q1q2 sources; 4 points from mentioned 4q1q2 periodic ones are fixed, and the remaining 4q1q2−4 points have period 2. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism f of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over f. The authors introduce the concept of suspension over diffeomorphisms of class G, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold T2×[0,1]/φ, where φ:T2→T2. The main result of the paper says that suspensions over diffeomorphisms of the class G are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions ϕt and ϕ′t implies the topological conjugacy of ϕ and ϕ′.