A Relativised Variational Principle for Continuous Transformations

Measures of maximal relative entropy

“…The theory of pressure and equilibrium states (see [16,6,7]), relative pressure and relative equilibrium states [8,20], and compensation functions [2,20] provides basic tools in this area. For a factor map π : X → Y between compact topological dynamical systems and potential function V ∈ C(X), Ledrappier and Walters [8] Any measure µ that attains the supremum is called a relative equilibrium state. A consequence is that the ergodic measures µ that have maximal entropy among all measures in π −1 {ν} have relative entropy given by…”

Algebraic $\mathbb {Z}^d$-actions of entropy rank one

“…The measure entropy of such a skew product has a simple expression in terms the Lyapunov vectors of the action. Using the relative variational principle of Ledrappier and Walters [17], we then show that the topological entropy of a continuous skew product is the largest of a finite number of topological pressures, analogous to the result of Marcus and Newhouse [21,Thm. B].…”

Variational Principle for Relative Tail Pressure

“…We need the following result, which has appeared already at several places (see for instance [8,10]). …”

“…A relative version of the variational principle for topological pressure was given by Ledrappier and Walters [8] in the framework of the relativized ergodic theory, and it was extended by Bogenschütz [9] to random transformations acting on one place. Later, Kifer [10] gave the variational principle for random bundle transformations.…”