2 problems

Smooth Markov partition conjecture. The induced toral endomorphism $f$ admits a smooth, in fact linear, Markov partition if and only if for some $k\in\mathbb{N}$, $A^k$ is diagonal…

Let $M_{\mathrm{A}}=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)\in\operatorname{SL}(2,\mathbb{Z})$, and let $f_0=0$, $f_1=1$, and $f_{k+1}=f_k+f_{k-1}$ for…