Scaled pressure of dynamical systems

A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles

“…the following result provides an exact formula for the upper metric mean dimension mdim M (Y N , d ρ , σ, ϕ), yielding a reformulation of the variational principle (7) in this setting.…”

“…In addition, if τ : P a (X) → [0, +∞] is another function taking the role of M in the equality (7), then τ ≤ M . Moreover, C Γ ⊇ lim inf ε → 0 + A Γε , where…”

“…Let (X, d) be a compact metric space, T : X → X be a continuous map such that mdim M (X, d, T ) < +∞ and ϕ ∈ C 0 (X) be a continuous potential. Then any equilibrium state µ ϕ of ϕ with respect to the variational principle (7) is invariant under T.…”

“…It was proved in [7] that, for every δ ∈ ]0, 1[ and every non-negative ϕ ∈ C 0 (X), mdim M (X, d, T, ϕ) = lim sup…”

“…[11]), just to mention a few. Regarding variational principles for continuous potentials and the previous measuretheoretic notions, we refer the reader to [7]. We stress that in all these variational principles the maximization on the space of probability measures is done for a fixed scale, which only afterwards is made to decrease to zero.…”