Let $X \subset \{0,1\}^{\mathbb{N}}$ be a compact, infinite, shift invariant set. Does $X$ contains a non-periodic point?.
My attempt: I was trying to construct a non-periodic point, given that we have orbits of randomly large periods inside $X$, but without arriving at anything interesting. My second idea was to construct somehow a open family for $X$, use the fact that $X$ is compact and extract a finite subfamily to conclude that the period of any point in $X$ is bounded by some constant, arriving to a contradiction ($X$ must be infinite)