Free Energy and Equilibrium States for Families of Interval Maps

Equilibrium States for Interval Maps: Potentials

We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition supφ − inf φ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of PerronFrobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important ...

Statistical Stability of Equilibrium States for Interval Maps

We consider families of transitive multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential φt : x 7→ −t log |Df(x)|, for t close to 1. We show that these equilibrium states vary continuously in the weak topology within such families. Moreover, in the...

Equilibrium States for Interval Maps: Potentials of Bounded Range

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition supφ − inf φ < htop, first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of PerronFrobenius operators. We demonstrate that this ‘bounded range’ condition on the potential i...

Equilibrium States for Factor Maps between Subshifts

Let π : X → Y be a factor map, where (X,σX) and (Y, σY ) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a = (a1, a2) ∈ R with a1 > 0 and a2 ≥ 0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes ∫ f dμ+ a1hμ(σX) + a2hμ◦π−1(σY ). In pa...

Equilibrium States for S-unimodal Maps