Measure Pressure for Measure Preserving Maps and an Upper Bound for the Case of \(C^2\) Endomorphisms

A measure-theoretic pressure was defined by [L. He, J. Lv and L. Zhou: Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica (English Series) 20, 709–718 (2004)] based on the Katok entropy formula. For a measure preserving map f, we generalized this definition to define a measure-theoretic pressure by using both \((n,\epsilon )\)-spanning and \((n,\epsilon )\)-separated sets. A variational principle for this pressure is established. Furthermore, we investigate an upper bound for the measure theoretic pressure of a \(C^{2}\) endomorphism preserving a hyperbolic measure.