Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators
This paper generalizes previous results to any Parry polynomials $p_n(z)$ \cite{Pa} of finite degree $n\geq 2$. Let $β>1$ be the unique positive root of $p_n$ and consider the transfer operator $\mathscr{P}$ induced by the map $[0,1]\ni x\mapsto βx-\lfloor βx\rfloor \in [0,1)$. Suppose all other roots of $p_n$ have magnitude less than one. Then for any smooth function $F$ on $[0,1]$ we prove two term asymptotics in $L^\infty$ of the form \begin{equation*} \big(\mathscr{P}^r(F)\big)(\cdot)= u(\cdot) \int_0^1 F(x)\: \mathrm{d} x +β^{-r}v(\cdot) \big (F(1)-F(0)\big )+o(β^{-r}),\quad r\geq 1, \end{equation*} where $u$ is piecewise constant and $\mathscr{P}u=u$, while $v$ is piecewise equal to a polynomial of degree one and $\mathscr{P}v= β^{-1}v$. When $\mathscr{P}$ is restricted to functions of bounded variation $BV([0,1])$, it is known that $\vert z \vert\leq β^{-1}$ contains its essential spectrum. Thus the contribution coming from $v$ in the above estimate is not guaranteed if $F$ is not smooth enough. Furthermore, we show that when $\mathscr{P}$ is restricted to $BV([0,1])$ then any $\vert z \vert<β^{-1}$ is an eigenvalue with infinite multiplicity. Moreover, we construct a meromorphic extension of $(\mathscr{P}-z)^{-1}$ to the domain $\vert z \vert>β^{-1}$, where the finite rank poles are numbers $ζ$ with $β^{-1}<\vert ζ\vert\leq 1$ such that $βζ$ is a root of $p_n(z)$. If such poles $ζ$ other than $1$ exist, then we construct eigenvectors of $\mathscr{P}^*$ in $BV^*$ obeying $\mathscr{P}^*\hat{U}=ζ\hat{U}$, where $\hat{U}(F)$ is given by series involving the values of $F$ taken in some specially chosen, finer and finer partitions of $[0,1]$. When $F$ is either a polynomial or a characteristic function like $χ_{[0,x]}$, then $\hat{U}(F)$ can be explicitly computed.