Dynamical Systems Seminar

A basic sequence (a sequence of bases) Q := (q_n)_{n = 1}^\infty is an element of \mathbb{N}_{\ge 2}^\mathbb{N}. Each basic sequence Q comes with an associated Cantor series expansion for all x \in [0,1] given by x = \sum_{n = 1}^\infty\frac{x_n}{\prod_{m = 1}^nq_m} = \frac{x_1}{q_1}+\frac{x_2}{q_1q_2}+\frac{x_3}{q_1q_2q_3}+\cdots, with x_n \in [0,q_n). While there are many numeration systems with a well developed theory of normality, such as base b, continued fractions, and beta expansions, an arbitrary basic sequence Q need not have such a theory for its Cantor series. In this talk we will consider classes of basic sequences Q that are generated by dynamical systems and show that they have a cohesive theory of normality.