(Variational principle) Sample Clauses

(Variational principle). For any H¨older continuous function f, P (f ) = sup .h(µ) + ∫ fdµΣ (2.5) where the supremum is taken over all σ-invariant probability measures. We say that an invariant measure µ is an equilibrium state if it achieves the supremum in (2.5), that is, P (f ) = h(µ) + ∫ fdµ. Recall the definition of a Xxxxx measure given in (2.1). An important result that will be used in this thesis is that, given a H¨older continuous potential f , there exists a unique invariant Xxxxx measure that can be characterised using the pressure and variational principle.

(Variational principle). If f is locally H¨older then

(Variational principle) αβ The variational principle with these boundary conditions requires an additional boundary term on top of the Xxxxxxx-Xxxxxxx term (2.48) and cosmological coun- terterm (2.49) of the Xxxxx-Xxxxxxxx story. This is due to the fact that, unlike with Xxxxx-Xxxxxxxx, we do not have δγ(0) = 0 for any α, β, and so the surface term (2.50) does not vanish. Explicitly, (2.50) with the CSS boundary conditions (2.65) gives: δ S + SGH + SCC |CSS = − 16πℓG ∫ d2x√−γ(0)γ(2)αβδγ(0) , αβ