Higher-dimensional maps with infinitely many Smale horseshoes

A class of higher-dimensional (invertible or non-invertible) maps with infinitely many Smale horseshoes is introduced, where the map is topologically conjugate to a symbolic dynamical system on each horseshoe, the Lebesgue measure of the parameters with the coexistence of infinitely many Smale horseshoes is infinite, and the topological entropy is infinite in certain parameter regions. Furthermore, these higher-dimensional maps are introduced to illustrate a geometric difference between horseshoes on plane and in Rn (n≥3), which is described by an index, denoted by (n,q,l), where 1≤l≤q≤n−1, that is, on each horseshoe, there exists a uniformly hyperbolic invariant set such that: (i) the maps of this type restricted to the invariant set are topologically conjugate to the two-sided full-shift on 2l symbols; (ii) the dimension of the unstable manifold of the invariant set is q, and dimension of the stable manifold is n−q.