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

We consider base-$\beta$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ 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 $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=\beta^{-1} v$ 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 $\mathscr{P}^kF= u +\beta^{-k}\big ( F(1)-F(0)\big )v +o(\beta^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.