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

### Georg Lohöfer

RWTH Aachen, Germany### Dieter Mayer

RWTH Aachen, Germany

## Abstract

An obviously little known result by Florek and Slater about the exact recurrence times of the sequence *mβ* mod 1 with respect to an arbitrary connected interval *I* in the unit interval is generalized to disconnected intervals *Ia,b* = [0, *a*)∪(*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 *Ia,b*. The relation of this result to the recurrence properties of integrable Hamiltonian systems with two degrees of freedom is obvious.

## Cite this article

Georg Lohöfer, Dieter Mayer, On a Theorem by Florek and Slater on Recurrence Properties of Circle Maps. Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, pp. 335–357

DOI 10.2977/PRIMS/1195171083