## The $$\beta$$-transformation with a hole (Joint Discrete-Dynamics Seminar)

#### Lyndsey Clark (University Of Manchester, School of Mathematics)

Let $$\beta \in (1, 2)$$ and consider the $$\beta$$-transformation
$T_\beta(x)=\beta x~ (\text{mod } 1).$
Take some interval $$(a, b) \subset [0, 1)$$ and call this a hole. Then we study the set of points whose orbits do not fall into the hole:
$\mathcal J_\beta (a,b) := \{ x \in (0,1) : T^n_\beta(x)\notin (a,b)\text{ for all }n\geq 0\}.$
If the hole $$(a, b)$$ is large, then ‘most’ orbits should fall in and so $$\mathcal J_\beta (a,b)$$ should be small, and vice versa. In this talk we will use symbolic dynamics and combinatorics on words to describe precisely the relationship between the hole $$(a, b)$$ and the size of $$\mathcal J_\beta (a,b)$$.