Upper Box Dimension Estimation For An Arbitrary Subset Of The Non--conformal Repeller

The topological pressure is a significant notion in the thermodynamic formalism and is an important method to study the dimension of sets.Using classical thermo-dynamic formalism,the dimension of a conformal repeller is obtained.However,the dimension estimates in non-conformal setting is much more difficult because of the limitations of the classical thermodynamic formalism.With further development of the dimension theory in dynamical system,it is found that the dimension of a non-conformal repeller can be well estimated by the root of non-additive topological pressure.In this paper,the Box dimension of an arbitrary subset of a non-conformal repeller is consid-ered.By studying the upper capacity topological pressure of the singular value function,we give an upper bound on the Box dimension of an arbitrary subset for a non-conformal repeller.This upper bound is the solution to some pressure function.Our innovation is that we do not need to assume the subset to be compact or invariant.Our argument ex-tends a series of topological pressure theories which are usually established on compact invariant sets to subsets that do not have to be compact or invariant.