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For $\xi \in \big( 0, \frac{1}{2} \big)$, let $E_{\xi}$ be the perfectsymmetric set associated with $\xi$, that is $$E_{\xi} = \Big\{ \exp \Big( 2i\pi (1-\xi) \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \Big) : \,\epsilon_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\}$$ and $$b(\xi) =\frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}.$$ Let$q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that anyinvertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big),\,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big(e^{n^{\beta}} \big), \, n \rightarrow +\infty \textrm{ for some } \beta <b(1/q),\end{eqnarray*} also satisfies the stronger property $\big\| T^{-n}\big\| = O \big( n^{s} \big), \, n \rightarrow +\infty.$ We also show that thisresult is false for $E_\xi$ when $1/\xi$ is not a Pisot number and that theconstant $b(1/q)$ is sharp. As a consequence we prove that, if $\omega$ is asubmulticative weight such that $\omega(n)=(1+n)^s, \, (n \geq 0)$ and $C^{-1}(1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},\, (n\geq 0)$, for someconstants $C>0$ and $\beta < b( 1/q),$ then $E_{1/q}$ satisfies spectralsynthesis in the Beurling algebra of all continuous functions $f$ on the unitcircle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) |\omega (n) < +\infty$.

On powers of operators with spectrum in cantor sets and spectral synthesis