Eulerian entropy and non-repetitive subword complexity

Abstract

We consider continuous self-maps of compact metric spaces, and for each point of the space we define the notion of eulerian entropy by considering the exponential growth rate of complexity in the initial chunks of the orbit of the point. We show that eulerian entropy is constant on a residual subset for transitive dynamical systems. For elements in the shift dynamical system we define an equivalent notion named non-repetitive subword complexity, and show that for a large class of mixing subshifts of finite type, the set of points for which the non-repetitive subword complexity is equal to the topological entropy is residual. If f is either a transitive interval map or an infinite transitive subshift of finite type, we establish that there is t∈N such that the eulerian entropy of ft is a positive constant that is attained on a residual set of points. © 2011 Elsevier B.V. All rights reserved.