We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component a limit measure will have at least that energies. From this, we deduce results concerning existence continuity equilibrium states (including statistical stability). Metric...

Given a compact topological dynamical system $(X, f)$ with positive entropy and upper semicontinuous map, any closed invariant subset $Y \subset X$ entropy, we show that there exists continuous roof function such the set o

4. Let (Ω,Σ, μ) be a measure space. If f : Ω → R, then we say that f is a measurable function if for every t ∈ R, the level set: Sf (t) = {x ∈ Ω|f(x) > t} is measurable. f : Ω→ C is measurable if and only if both Re(f) and Im(f) are measurable. 5. A real function f : Ω → R is lower semicontinuous on Ω if Sf (t) is open, and is upper semicontinuous if {x ∈ Ω|f(x) < t} is open. Equivalently, f is...