# Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type

@article{Jenkinson2005ZeroTL, title={Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type}, author={Oliver Jenkinson and R. Daniel Mauldin and Mariusz Urbanski}, journal={Journal of Statistical Physics}, year={2005}, volume={119}, pages={765-776} }

Let ΣA be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f:ΣA → IR, the family of Gibbs-equilibrium states (μtf)t⩾1 for the functions tf is shown to be tight. Any weak*-accumulation point as t→∞ is shown to be a maximizing measure for f.

## 53 Citations

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## References

SHOWING 1-10 OF 35 REFERENCES

Ergodic optimization for countable alphabet subshifts of finite type

- MathematicsErgodic Theory and Dynamical Systems
- 2006

Let $X$ be a one-sided subshift of finite type on a countable alphabet, and $T:X\to X$ the shift map. If $f:X\to{\mathbb R}$ is continuous, we provide conditions guaranteeing that $f$-maximizing…

Gibbs measures at temperature zero

- Mathematics
- 2003

Let νf be the Gibbs measure associated with a regular function f on a one-sided topologically mixing subshift of finite type. Introducing a parameter λ, we consider the behaviour of the family (νλf),…

Existence of gibbs measures for countable Markov shifts

- Mathematics
- 2003

We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and…

Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics

- Mathematics
- 1990

We study various aspects of the dynamics of skew-products (Rand Z-extensions) over a subshift of finite type (ssft).
In Chapter I we give the basic definitions and terminology.
Conditions are…

COUNTABLE EXTREME GIBBS STATES IN A ONE-DIMENSIONAL MODEL WITH UNIQUE GROUND STATE

- Mathematics
- 1998

A one-dimensional model having a unique ground state and countable number of extreme limit Gibbs states is constructed.

Introduction to Ergodic Theory

- Mathematics
- 1977

Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time…

Disordered Ground States of Classical Lattice Models

- Mathematics
- 1991

We use strictly ergodic dynamical systems to describe two methods for constructing short range interactions of classical statistical mechanics models with unique ground states and unusual properties…

Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps

- MathematicsErgodic Theory and Dynamical Systems
- 2003

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps…

Ergodic optimization for noncompact dynamical systems

- Mathematics
- 2007

The purpose of this note is to initiate the study of ergodic optimization for general topological dynamical systems T:X→ X, where the topological space X need not be compact. Given , four possible…

ERGODIC OPTIMIZATION FOR NON-COMPACT DYNAMICAL SYSTEMS

- Mathematics
- 2003

The purpose of this note is to initiate the study of ergodic optimization for general topological dynamical systems T : X → X, where the topological space X need not be compact. Given f : X → R, four…