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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 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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