No abstract

We consider dynamical systems (X, T, µ) which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls (Bn) ∞ n=1 , we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a k < m with T k (x) ∈ Bm. We also give an asymptotic estimate as m → ∞ on the number of k < m with T k (x) ∈ Bm.As an application, we prove for almost every point x an asymptotic estimate on the number of k ≤ m such that a k ≥ m t , where t ∈ (0, 1) and a k are the continued fraction coefficients of x.