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

For $\xi \in \big(0, {1/2} \big)$, we denote by $E_{\xi}$ the perfect symmetric set associated to $\xi$, that is $$E_{\xi} = \Big\{\exp \big(2i \pi (1-\xi) \dsp \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \big) : \epsilon_{n} = 0 \textrm{or} 1 \quad (n \geq 1) \Big\}.$$ Let $s$ be a nonnegative real number, and $T$ be an invertible bounded operator on a Banach space with spectrum included in $E_{\xi}$. We show that if \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big(n^{s} \big), n \to +\infty & \textrm{and} & \big\| T^{-n} \big\| = O \big(e^{n^{\beta}} \big), n \to +\infty \textrm{for some} \beta < \frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}, \end{eqnarray*} then for every $\e > 0$, $T$ satisfies the stronger property $$\big\| T^{-n} \big\| = O \big(n^{s+{1/2}+\e} \big), n \to +\infty.$$ This result is a particular case of a more general result concerning operators with spectrum satisfying some geometrical conditions.