Let T be an expanding Markov map with a repeller F on \(X\subset [0,1]\) . For any \(\{y_n\}_{n\ge 0}\subset [0,1]\) , we prove that the Hausdorff dimension of the badly approximable set \(\begin{aligned} \Big \{x\in F: \liminf \limits _{n\rightarrow \infty }|T^n (x)-y_n|>0 \Big \} \end{aligned}\) is full. Moreover, using this result, we conclude that the Hausdorff dimension of the set of non-recurrent points, defined by \(\begin{aligned} \Big \{x\in F: \liminf \limits _{n\rightarrow \infty }|T^n(x)-x|>0 \Big \} \end{aligned}\) is also full. The results can be applied to some dynamical systems on fractal sets, such as cookie-cutter systems.
