Equilibrium State for One-Dimensional Lorenz-Like Expanding Maps

Abstract

Let L : [0, 1] \ {d} → [0, 1] be a one-dimensional Lorenz like expanding map (d is the point of discontinuity), P = {(0, d), (d, 1)} be a partition of [0, 1] and C α ([0, 1], P) the set of piecewise Hölder-continuous potential of [0,1] with the usual C 0 topology. In this context, we prove, improving a result of [2], that piecewise Hölder-continuous potential φ satisfying max n lim sup n→∞ 1 n (S n φ)(0), lim sup n→∞ 1 n (S n φ)(1) o < P top (φ, T) support an unique equilibrium state. Indeed, we prove there exists an open and dense subset H of C α ([0, 1], P) such that, if φ ∈ H, then φ admits one equilibrium measure.Contents 14 Proof of that H is C 0 -dense in C α ([0, 1], P) 16 References 17

Phase Transitions for one-dimensional Lorenz-like expanding Maps

“…In order to find a periodic point for an one-dimensional Lorenz-like expanding map we define C + n and C − n as the cylinders on the right and left hand side of the discontinuity d, respectively, i.e., d ∈ ∂C ± n . For this purpose we introduce an auxiliary family A n by recursively as follows: let [3,10]) An integer N is a cutting time for T if d ∈ A N .…”

“…Lemma 2.3. (see [3]) Let ℓ : [0, 1] \ {d} → [0, 1] be an one-dimensional Lorenz-like expanding map and consider φ ∈ C α ([0, 1], P).…”

“…Proof. We Just use Lemma 2.2 and make the superficial modifications to the proof of Proposition 3.1 of [3].…”

“…Such studies can be directed toward understanding and discovering dynamic properties from both the topological and ergodic points of view, which are satisfied for a "large" set of dense or residual systems. In this context, Bronzi and Oler [3] proved that if ℓ is an one-dimensional Lorenz-like expanding map and H α ([0, 1], R) is the set of piecewise H ölder-continuous potentials of [0,1] with the usual C 0 topology, then there exists an open and dense subset H of H α ([0, 1], R) such that each φ ∈ H admits exactly one equilibrium state. The next theorem can be seen as a generalization of this result.…”

“…In [3] potentials no presenting property (1.1) were not studied. However, the authors proved that if φ is a piecewise H ölder-continuous potential and max lim sup n→∞ 1 n (S n φ)(0), lim sup n→∞ 1 n (S n φ)(1) is an upper bound for topological pressure then φ admits a unique equilibrium measure.…”