Sharp iteration asymptotics for transfer operators induced by greedy $\beta$-expansions

We consider base-$\beta$ expansions of Parry’s type, where $a_0\ge a_1\ge 1$ are integers and $a_0<\beta <a_0+1$ is the positive solution to ${\beta }^2=a_0\beta +a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto \beta x-\lfloor \beta x\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathcal{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathcal{P}u=u$ and $\mathcal{P}v={\beta }^{-1}v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies ${\int }_0^{1}F\mathrm{d}x=1$, we have ${\mathcal{P}}^{k}F=u+{\beta }^{-k}(F(1)-F(0))v+o({\beta }^{-k})$ in $L^{\mathrm{\infty }}$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.