Spectral and dynamical results related to certain non-integer base expansions on the unit interval

We consider certain non-integer base $\beta$-expansions of Parry’s type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\ge 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $\beta$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{{\mathcal{P}}^{N}f{\}}_{N\ge 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This “attracting” eigenvalue is not isolated: for $1\le p\le 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions.