Let (X, B, µ, T, d) be a measure-preserving dynamical system with exponentially mixing property, and let µ be an Ahlfors s-regular probability measure. The dynamical covering problem concerns the set E(x) of points which are covered by the orbits of x ∈ X infinitely many times. We prove that the Hausdorff dimension of the intersection of E(x) and any regular fractal G equals dim H G + α − s, where α = dim H E(x) µ-a.e. Moreover, we obtain the packing dimension of E(x) ∩ G and an estimate for dim H (E(x) ∩ G) for any analytic set G.

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Let [Formula: see text] be a limsup random fractal with indices [Formula: see text] and [Formula: see text] on [Formula: see text]. We determine the hitting probability [Formula: see text] for any analytic set [Formula: see text] with the condition [Formula: see text], where [Formula: see text] denotes the Hausdorff dimension. This extends the correspondence of Khoshnevisan et al.1 by relaxing the condition that the probability [Formula: see text] of choosing each dyadic hyper-cube is homogeneous and [Formula: see text] exists. We also present some counterexamples to show the Hausdorff dimension in condition [Formula: see text] cannot be replaced by the packing dimension.