Christian Roettger: Periodic points in Markov shifts
- or: meet the world's two weirdest primes!

Let G be a finite group and X the space of doubly indexed sequences x(s,t) over G satisfying the condition
X carries two shift actions (left, down). This gives a natural generalization of the notion of periodic points. For a subgroup H of Z2 of finite index, we say that a sequence in X is H-periodic if it is invariant under all movements h in H. Let FH be the number of H-periodic sequences in X. Tom Ward (Norwich, UK) has shown that the numbers FH determine the underlying group G up to isomorphism in most cases. Only powers of 2 and the so-called Wieferich primes cause trouble (as they often do). Wieferich primes are primes p such that 2(p-1)-1 is divisible by p2. We propose a simple extension of his ideas that could settle the remaining cases. This involves doing 'algebraic geometry' with the space X as a Z[S,T]-module and studying ideals in Z/(pn)[T]. We can use much of Ruben Aydinyan's talk.

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