Sharp iteration asymptotics for transfer operators induced by greedy ${\hat{I}}^2$-expansions

Abstract

We consider base-${\hat{I}}^2$ expansions of Parry's type, where $a_0\ge a_1\ge 1$ are integers and $a_0<{\hat{I}}^2<a_0+1$ is the positive solution to $β^2 = a_0β+ a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto {\hat{I}}^{2}x-\lfloor {\hat{I}}^{2}x\rfloor$ 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 $\mathscr{P}v=β^{-1} v$ 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 $\mathscr{P}^kF= u +β^{-k}\big ( F(1)-F(0)\big )v +o(β^{-k})$ in $L^{\infty }$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

19 pages, 2 figures