Stability of symbolic embeddings for difference equations and their multidimensional perturbations

Abstract

In this paper, we study complexity of solutions of a high-dimensional difference equation of the form Phi(x(i-m), . . . , x(i-1), x(i), x(i+1), . . . , x(i+n)) = 0, i is an element of Z, where Phi is a C-1 function from (R-l)(m+n+1) to R-l. Our main result provides a sufficient condition for any sufficiently small C-1 perturbation of Phi to have symbolic embedding, that is, to possess a closed set of solutions Lambda that is invariant under the shift map, such that the restriction of the shift map to Lambda is topologically conjugate to a subshift of finite type. The sufficient condition can be easily verified when Phi depends on few variables, including the logistic and Henon families. To prove the result, we establish a global version of the implicit function theorem for perturbed equations. The proof of the main result is based on the Brouwer fixed point theorem, and the proof of the global implicit function theorem is based on the contraction mapping principle and other ingredients. Our novel approach extends results in [2,3,8,15,21]. (C) 2014 Elsevier Inc. All rights reserved.