# Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < htop(f)

@article{Bruin2008EquilibriumSF,
title={Equilibrium States for Interval Maps: Potentials with sup $\phi$ − inf $\phi$ < htop(f)},
author={Henk Bruin and Mike Todd},
journal={Communications in Mathematical Physics},
year={2008},
volume={283},
pages={579-611}
}
• Published 7 August 2008
• Mathematics
• Communications in Mathematical Physics
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 sup φ − inf φ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this…
The statistical stability of equilibrium states for interval maps
• Mathematics
• 2009
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
A characterization of hyperbolic potentials of rational maps
• Mathematics
• 2011
Consider a rational map f of degree at least 2 acting on its Julia set J(f), a Hölder continuous potential φ: J(f) → ℝ and the pressure P(f,φ). In the case where \mathop {\sup }\limits_{J(f)} \phi
Natural Equilibrium States for Multimodal Maps
• Mathematics
• 2010
This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove
Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials
• Mathematics
• 2012
There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map
Thermodynamic formalism for expanding measures
• Mathematics
• 2022
We study the Thermodynamic Formalism for strongly transitive endomorphisms f , focusing on the set all expanding measures. When f is a non-flat C map defined on a Riemannian manifold, being an
Multifractal analysis for multimodal maps
Given a multimodal interval map $f:I \to I$ and a H\"older potential $\phi:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $\phi$. The main tool here is inducing schemes,
Equilibrium measures for maps with inducing schemes
• Mathematics
• 2008
We introducea class of continuousmaps f of a compact topologi- cal space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a

## References

SHOWING 1-10 OF 76 REFERENCES
Large derivatives, backward contraction and invariant densities for interval maps
• Mathematics
• 2008
In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfn(f(c))|→∞ as n→∞ holds for all
Equilibrium states for piecewise monotonic transformations
• Physics, Mathematics
Ergodic Theory and Dynamical Systems
• 1982
Abstract We show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases: (i) sup φ — inf φ <htop(T) and φ
Equilibrium states for interval maps: the potential $-t\log |Df|$
• Mathematics
• 2007
Let $f:I \to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential
Pressure and equilibrium states for countable state markov shifts
• Mathematics
• 2002
We give a general definition of the topological pressurePtop(f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for
Ergodic theory of equilibrium states for rational maps
• Mathematics
• 1991
Let T be a rational map of degree d>or=2 of the Riemann sphere C=C union ( infinity ). The authors develop the theory of equilibrium states for the class of Holder continuous functions f for which
Examples for the nonuniqueness of the equilibrium state
In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the
Equilibrium states for S-unimodal maps
• Physics
Ergodic Theory and Dynamical Systems
• 1998
For S-unimodal maps $f$, we study equilibrium states maximizing the free energies $F_t(\mu) := h(\mu) - t\int \log|f'|\,d\mu$ and the pressure function $P(t):=\sup_\mu F_t(\mu)$. It is shown that if
Lifting measures to Markov extensions
Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we
Specification on the interval
We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to
Markov extensions for multi-dimensional dynamical systems
• J. Buzzi
• Mathematics, Computer Science
• 1999
This work generalizes the result of F. Hofbauer on piecewise monotonic maps of the interval to arbitrary piecewise invertible dynamical systems and gets a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy.