A relative local variational principle for topological pressure

Abstract

We define the relative local topological pressure for any given factor map and open cover, and prove the relative local variational principle of this pressure. More precisely, for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover U of X, a continuous, real-valued function f on X and an S-invariant measure ν on Y, we show that the corresponding relative local pressure P(T, f, U, y) satisfies $$ \mathop {\sup }\limits_{\mu \in \mathcal{M}(X,T)} \left\{ {h_\mu (T,\mathcal{U}|Y) + \int_X {f(x)d\mu (x):\pi \mu = \nu } } \right\} = \int_Y {P(T,f,\mathcal{U},y)d\nu (y)} , $$ where \( \mathcal{M} \)(X, T) denotes the family of all T-invariant measures on X. Moreover, the supremum can be attained by a T-invariant measure.