Hausdorff dimensions of recurrent and shrinking target sets under Lipschitz functions for expanding Markov maps

Let T be an expanding Markov map with a repeller E defined on X subset of [0, 1]. This paper concerns the Hausdorff dimension of the sets {x is an element of X : |T(n)x - g(n)(x)| < e(-Snf(x)) for infinitely many n is an element of N} and {x is an element of X : |T(n)x - g(n)(x)| < psi(n) for infinitely many n is an element of N}, where {g(n)}(n >= 0) is a sequence of Lipschitz functions with a uniform Lipschitz constant, g(n) : X -> (E) over bar, f is a positive continuous function on [0, 1], S(n)f (x) is the sum f (x) + f (Tx) + f (T(2)x)+ center dot center dot center dot + f (T(n-1)x) and psi is a positive function defined on N. The results can be applied to cookie-cutter dynamical systems and continued fraction dynamical systems, etc.