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Existence of the zero-temperature limit of equilibrium states on topologically transitive countable Markov shifts
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 04 October 2022, pp. 3231-3254
- Print publication:
- October 2023
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Consider a topologically transitive countable Markov shift
$\Sigma $ and a summable locally constant potential 
$\phi $ with finite Gurevich pressure and 
$\mathrm {Var}_1(\phi ) < \infty $. We prove the existence of the limit 
$\lim _{t \to \infty } \mu _t$ in the weak
$^\star $ topology, where 
$\mu _t$ is the unique equilibrium state associated to the potential 
$t\phi $. In addition, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.
