# Some free actions for one-sided shift spaces

### Collin Bleak (University of St Andrews)

Thursday 27th October, 2022 16:00-17:00 Maths 311B

#### Abstract

Boyle and Krieger in 1987 provide an invariant of conjugacy for the group Aut($\{0, 1, ..., n-1\}^N, \sigma$) of automorphisms of the full one-sided shift on alphabet {0, 1, ..., n-1}. The invariant arises from the action of the group on finite periodic words and consists of a tuple: the well-known gyration and sign functions, together with bundled-first-return data (bundled data associated to the permutation representation on prime words of length k for each k). If two automorphisms are not conjugate, then in finite time you can detect this using the invariant. However, while these invariants can be constructed from given automorphisms, finding an automorphism with given data is not an easy task.

Boyle has proposed the following question: Consider the full (one-sided) shift on n-letters. Does there exists an automorphism A of order n acting freely on the shift space (every point has an orbit of length n under the action of the automorphism), but where A is not conjugate to a (cyclic) permutation on n letters? In this talk, we show there is an algorithm to reduce any such automorphism, through conjugacy, to a cyclic permutation. Joint with Feyisayo Olukoya.