The dimensions of a non-conformal repeller and an average conformal repeller

In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C¹, without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.