Abstract. We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let b ≥ 2 be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of b-ary digits is a Sturmian sequence over {0, 1, . . . , b − 1} and we prove that this lower bound is best possible. As an application, we derive some information on the b-ary expansion of log(1 + 1 a ), for any integer a ≥ 34.
We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.
We investigate the recurrence property of irrational rotations. Let T be the rotation by an irrational θ on the unit circle. We show that for a fixed y lim infThis result is a metric inhomogeneous Diophantine approximation in an almost everywhere sense.
We consider the recurrence time to the r-neighbourhood for interval exchange maps (i.e.m.s). For almost every i.e.m. we show that the logarithm of the recurrence time normalized by − log r goes to 1. A similar result of the hitting time also holds for almost every i.e.m.
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