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 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.