The spectrum of Poincare recurrence

We investigate the relationship between Poincare recurrence and topological entropy of a dynamical system (X, f). For 0 <= alpha <= beta <= infinity, let D(alpha, beta) be the set of x with lower and upper recurrence rates a and P, respectively. Under the assumptions that the system is not minimal and that the map f is positively expansive and satisfies the specification condition, we show that for any open subset empty set not equal U subset of X, D(alpha, beta) boolean AND U has the full topological entropy of X. This extends a result of Feng and Wu [The Hausdorff dimension of recurrence sets in symbolic spaces. Nonlinearity 14 (2001), 81-85] for symbolic spaces.