Cycles in $C^ r$-twists

Abstract:

Let ${f_t},0 \leq t \leq 1$, be a continuous one-parameter family of ${C^r}$ diffeomorphisms of the circle obtained by monotonically twisting away from ${f_0}$. It is well known that for a dense set of parameter values ${t, f_t}$ has a periodic orbit. To what extent does the distribution in the circle of these periodic orbits reflect the degree of differentiability r? We show that if $r \geq 6,{\sup _t}{\left \| {{f_t} \circ f_0^{ - 1}} \right \|_{{C^r}}} < \infty$, and the rotation number of ${f_0}$ is an irrational $\alpha$ with bounded continued fraction expansion, then periodic orbits corresponding to small values of the parameter t echo the metric structure of ${f_0}$ in the following sense: If the rotation number of the orbit is a convergent of $\alpha$, then the orbit divides the circle into intervals of nearly equal $\mu$-measure, where $\mu$ is the invariant Borel probability measure of ${f_0}$. The corresponding result for low differentiability is false.