The \(\beta\)-transformation with a hole

Lyndsey Clark (University of Manchester)

Frank Adams Room 1, Alan Turing Building,

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 \(\mathcal J_\beta (a,b)\).
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