Stability of symbolic embeddings for difference equations and their multidimensional perturbations

In this paper, we study complexity of solutions of a high-dimensional difference equation of the formΦ(xi-m,...,xi-1,xi,xi+1,...,xi+n)=0,i∈Z, where Φ is a C¹ function from (Rℓ)m+n+1 to Rℓ. Our main result provides a sufficient condition for any sufficiently small C¹ perturbation of Φ to have symbolic embedding, that is, to possess a closed set of solutions Λ that is invariant under the shift map, such that the restriction of the shift map to Λ is topologically conjugate to a subshift of finite type. The sufficient condition can be easily verified when Φ depends on few variables, including the logistic and Hénon 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].