Abstract
We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity [Formula: see text] which describes the maximal length of string of zeros of the prefix among the first [Formula: see text] iterations of [Formula: see text] in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined.
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