# Differentiability of the pressure in non-compact spaces

@article{Iommi2020DifferentiabilityOT, title={Differentiability of the pressure in non-compact spaces}, author={Godofredo Iommi and Mike Todd}, journal={Fundamenta Mathematicae}, year={2020} }

Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property…

## 2 Citations

### Thermodynamical and spectral phase transition for local diffeomorphisms in the circle

- Mathematics
- 2021

It is known that all uniformly expanding dynamics have no phase transition with respect to Hölder continuous potentials. In this paper we show that given a local diffeomorphism f on the circle, that…

### Rigidity of pressures of H\"older potentials and the fitting of analytic functions via them

- Mathematics
- 2022

. The ﬁrst part of this work is devoted to the study of higher diﬀerentials of pressure functions of H¨older potentials on shift spaces of ﬁnite type. By describing the diﬀerentials of pressure…

## References

SHOWING 1-10 OF 60 REFERENCES

### A thermodynamic definition of topological pressure for non-compact sets

- MathematicsErgodic Theory and Dynamical Systems
- 2010

Abstract We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle,…

### Differentiability properties of the pressure in lattice systems

- Mathematics
- 1980

In two recent papers Ruelle gave a heuristic theory of phase transitions, using some techniques introduced by Israel. He proves a version of Gibbs phase rule, assuming a differentiability condition…

### Phase Transitions for Countable Markov Shifts

- Mathematics
- 2001

Abstract: We study the analyticity of the topological pressure for some one-parameter families of potentials on countable Markov shifts. We relate the non-analyticity of the pressure to changes in…

### Escape of entropy for countable Markov shifts

- Mathematics, Computer ScienceAdvances in Mathematics
- 2022

### Examples for the nonuniqueness of the equilibrium state

- Mathematics
- 1977

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…

### Thermodynamic Formalism for Transient Potential Functions

- MathematicsCommunications in Mathematical Physics
- 2019

We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and…

### Convex Functions, Monotone Operators and Differentiability

- Mathematics
- 1989

These notes start with an introduction to the differentiability of convex functions on Banach spaces, leading to the study of Asplund spaces and their intriguing relationship to monotone operators…

### Symbolic dynamics for non-uniformly hyperbolic systems

- MathematicsErgodic Theory and Dynamical Systems
- 2020

Abstract This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory…

### Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

- Mathematics
- 2000

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages…

### Canonical compactifications for Markov shifts

- Mathematics, PhysicsErgodic Theory and Dynamical Systems
- 2012

Abstract We give a complete characterization of the compact metric dynamical systems that appear as boundaries of the canonical compactification of a locally compact countable state mixing Markov…