Ergodic Optimization in the Expanding Case

Abstract

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Extremal norms for fiber-bunched cocycles

“…Important applications include [BMa, Con]. The study of invariant measures that attain that supremum in (1.7) is called ergodic optimization; we refer the reader to [Je1,Je2,Gar] for much more information.…”

“…In traditional ergodic optimization, that is, the optimization of Birkhoff averages (see [Je1,Je2,Gar]), a maximizing set is a closed subset such that an invariant probability is maximizing if and only if its support lies on this subset. The existence of such sets is guaranteed in any context where a Mañé Lemma holds.…”

Applications of variable discounting dynamic programming to iterated function systems and related problems

“…Following the classical approach in thermodynamical formalism as in [Bou01] and [LMMS15] we have that the Bellman and the discounted transfer operators are given by In [Bou01] and [LMMS15] the author shows that the discounted limit of the first one provides a calibrated subaction equation: It is well known (see [Bou01] or [Gar17]) that a measure µ max satisfying m ∞ = sup T * µ=µ X g dµ = X ϕ dµ max , is supported in {y ∈ X | v ∞ (T (y)) = ϕ(y) − m ∞ + v ∞ (y)}. It is also well known (see [Bou01] or [LMMS15]) that the discounted limit of the second one gives a positive eigenfunction e v∞(x) and an maximal eigenvalue e k (which is the spectral radius) of the Ruelle operator, that is,…”

Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property

“…For transitive expanding dynamics, generic continuous potentials do not admit bounded measurable sub-actions (see [BJ02, Theorem C] and for details [Gar17,Appendix]). Surprisingly there are few cases in the literature about specific examples of non-existence of continuous sub-actions.…”