Soficity, recurrence and short cycles of exponential maps (joint work with K. Juschenko)

Abstract
Let $f$ be an exponentiation map mod $p$, or, more precisely, the map from $\{0,1,...p-1\}$ to itself defined by $f(x) \equiv 2^x \mod p$. It is easy to show that $f$ and $f\circ f$ have few fixed points. Showing that $f\circ f\circ f$ has $o(p)$ fixed points is harder, and was open; we show how to prove it. What about $f\circ f\circ f\circ f$? There, the problem is still open; we show its connection to {\em sofic groups}. More precisely: if the Higman group is sofic, then there is a map $f$ that (a) behaves almost everywhere like an exponentially map, and (b) satisfies $f(f(f(f(x))))=x$ for almost all x. The proof rests in part on an elementary proof of a special case of the uniqueness of sofic representations of amenable groups.