Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

Sarah Day, Rafael M. Frongillo

Abstract: We extend and demonstrate the applicability of computational Conley index techniques for computing symbolic dynamics and corresponding lower bounds on topological entropy for discrete-time systems governed by maps. In particular, we describe an algorithm that uses Conley index information to construct sofic shifts that are topologically semiconjugate to the system under study. As an illustration, we present results for the two-dimensional Hénon map, the three-dimensional LPA map, and the infinite-dimensional Kot--Schaffer map. This approach significantly builds on methods first presented in [S. Day, R. Frongillo, and R. Treviño, SIAM J. Appl. Dyn. Syst., 7 (2008), pp. 1477--1506] and is related to work in [J. Kwapisz, Math. Z., 234 (2000), pp. 255--290; J. Kwapisz, Ergodic Theory Dynam. Systems, 24 (2004), pp. 1173--1197].