On the growth of powers of operators with spectrum contained in Cantor sets

For xi is an element of(0, 1/2), we denote by E-xi the perfect symmetric set associated to, that is, E xi = {exp (2i pi(1 - xi) (n=1)Sigma(+infinity) epsilon(n)xi(n-1)) : epsilon(n) = 0 or 1 ( n >= 1)}. Let s be a nonnegative real number, and T be an invertible bounded operator on a Banach space with spectrum included in E. We show that if parallel to T(n)parallel to = O(n(S)), n -> +infinity, parallel to T(-n)parallel to = O(e(n beta)), n -> +infinity for some beta < [log(1/xi) - log2]/[2log(1/xi) - log2] then for every epsilon > 0, T satisfies the stronger property parallel to T(-n)parallel to = O(n(s+1/2+epsilon)), n -> +infinity. This result is a particular case of a more general result concerning operators with spectrum satisfying some geometrical conditions.