Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved Climenhaga-Thompson criterion. (English) Zbl 1507.37042
The authors extend a theorem of V. Climenhaga and D. J. Thompson [Adv. Math. 303, 745–799 (2016; Zbl 1366.37084)]
concerning the existence and uniqueness of equilibrium states for flows with the purpose of applying the results to flows with fixed points.
Theorems of this flavor go back to the work of R. Bowen [Math. Syst. Theory 8, 193–202 (1975; Zbl 0299.54031)]
in the 1970’s where he considered an expansive discrete time dynamical system and gave sufficient regularity conditions for a potential to ensure that it has a unique equilibrium state. This was extended by E. Franco [Am. J. Math. 99, 486–514 (1977; Zbl 0368.54014)]
to the case of flows, and these results were then made applicable to much larger classes of systems by weakening the hypotheses, allowing bad behavior on small parts of the space (in terms of topological pressure).
The present paper continues this line of research by further weakening the hypotheses in the result of V. Climenhaga and D. J. Thompson [loc. cit.]. Specifically, the authors weaken the assumptions by requiring a regularity condition at a single scale rather than at all scales.
The present paper continues this line of research by further weakening the hypotheses in the result of V. Climenhaga and D. J. Thompson [loc. cit.]. Specifically, the authors weaken the assumptions by requiring a regularity condition at a single scale rather than at all scales.
Reviewer: Anthony Quas (Victoria)
MSC:
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
Keywords:
thermodynamic formalism; flows; equilibrium states; specification property; specification at a fixed scale; nonuniform dynamicsReferences:
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