On a Theorem by Florek and Slater on Recurrence Properties of Circle Maps

Abstract

An obviously little known result by Florek and Slater about the exact recurrence times of the sequence $m\beta \mathrm{mod}1$ with respect to an arbitrary connected interval $I$ in the unit interval is generalized to disconnected intervals $I_{\overgroup{a,b}}=[0,a)\cup (b,1)$ when $b=1-a$, $a<1/2$. It is shown that the formula of Florek and Slater expressing the possible recurrence times in terms of the interval $I$ is valid also in our case. This let us expect that this formula is valid also for general intervals of the form $I_{\overgroup{a,b}}$. The relation of this result to the recurrence properties of integrable Hamiltonian systems with two degrees of freedom is obvious.