On Eventually Always Hitting Points

We consider dynamical systems $$(X,T,\mu )$$ which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls $$(B_n)_{n=1}^\infty $$ , 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 $$T^k (x) \in B_m$$ . We also give an asymptotic estimate as $$m \rightarrow \infty $$ on the number of $$k < m$$ with $$T^k (x) \in B_m$$ . As an application, we prove for almost every point x an asymptotic estimate on the number of $$k \le m$$ such that $$a_k \ge m^t$$ , where $$t \in (0,1)$$ and $$a_k$$ are the continued fraction coefficients of x.