Topological pressure of proper map

Based on the Carathéodory-Pesin structure theory [13], we introduce three notions of topological pressure of a proper map and provide some properties of these notions. For the proper map of a locally compact separable metric space, we prove some variational principles and give some applications on the multifractal analysis of local entropies. These are the extensions of results of Pesin, Takens and Verbiski, etc.