The Hausdorff dimension of recurrent sets in symbolic spaces

Let (Sigma, sigma) be the one-sided shift space on m symbols. For any x = ( x(i))(i)greater than or equal to1 is an element of Sigma and positive integ er n, define R-n(X) = inf{j greater than or equal to n : x(1)x(2)...x(n) = x(j+l)x(j+2). ..x(j+n)}. We prove that for each pair of numbers alpha, beta is an element of [0, inf inity] with alpha less than or equal to beta, the following recurrent set E-alpha.beta={x is an element of Sigma : lim inf (n-->infinity) log R-n(x)/ n = alpha, lim sup(n-->infinity) log R-n(x)/n = beta} has Hausdorff dimension one. AMS classification scheme number: 28A80.