
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(s,t+1)=x(s,t)+x(s+1,t).
X carries two shift actions (left, down). This gives a natural
generalization of the notion of periodic points. For a subgroup H of Z^{2} of
finite index, we say that a sequence in X is Hperiodic if it is invariant
under all movements h in H.
Let F_{H} be the number of Hperiodic sequences in X.
Tom Ward (Norwich, UK) has shown that the numbers F_{H} determine the underlying
group G up to isomorphism in most cases. Only powers of 2 and the socalled
Wieferich primes cause trouble (as they often do).
Wieferich primes are primes p such that 2^{(p1)}1 is divisible by p^{2}.
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/(p^{n})[T]. We can use much of Ruben Aydinyan's talk.
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