Dynamical Collapse of Trajectories Part II: Limit Set of a Horseshoe-like Map

The dynamics and limit set of a discrete-time system is described which is similar to the horseshoe map of Smale. The particular characteristics of this map are motivated by the study of mechanical systems with dry friction, where, as is discussed in the companion paper [Biemond, J.J.B. et. al., Dynamical collapse of trajectories, part I: Homoclinic tangles in systems with dry friction, submitted to Proc. ENOC 2014], homoclinic orbits can exist that emanate from sets of non-isolated equilibrium points, and the dynamics near these orbits is described by a non-invertible return map. The dynamics of the horseshoe-like nonsmooth map is shown to be topologically conjugate to a symbolic dynamics. While trajectories of this map can be continued uniquely in the forward direction of time, only in a subset of the state space, a unique continuation exists for the trajectory in the backward time direction. The same property is introduced in the symbolic dynamics by defining this dynamics as a shift on a quotient space of the standard symbolic state space with infinite strings of two symbols. Using the mentioned conjugacy, it is proven that the limit set contains an infinite number of periodic orbits.