Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map.

Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid $$ S_f=\{\{x_n; n\in\mathbb Z\}: x_{n+1}=f(x_n)\} $$ equipped with the induced topology.

Question: Is $S_f$ contractible? If yes, is there a deformation retraction?

Motivation is here: Two commuting mappings in the disk