Large deviation principles for countable Markov shifts

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

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On the existence of maximizing measures for irreducible countable Markov shifts: a dynamical proof

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.194 ◽

2013 ◽

Vol 34 (4) ◽

pp. 1103-1115 ◽

Cited By ~ 5

Author(s):

RODRIGO BISSACOT ◽

RICARDO DOS SANTOS FREIRE

Keyword(s):

Probability Measure ◽

Bounded Variation ◽

Finite Alphabet ◽

Markov Shift ◽

Positive Real ◽

Shift Space ◽

Markov Shifts ◽

Maximizing Measure ◽

Full Shift ◽

Countable Markov Shifts

AbstractWe prove that if ${\Sigma }_{\mathbf{A} } ( \mathbb{N} )$ is an irreducible Markov shift space over $ \mathbb{N} $ and $f: {\Sigma }_{\mathbf{A} } ( \mathbb{N} )\rightarrow \mathbb{R} $ is coercive with bounded variation then there exists a maximizing probability measure for $f$, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case in the general irreducible non-compact setting. It is also noteworthy that our technique works for the full shift over positive real sequences.

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Large deviations and fluctuation theorem for selectively decoupled measures on shift spaces

Reviews in Mathematical Physics ◽

10.1142/s0129055x19500363 ◽

2019 ◽

Vol 31 (10) ◽

pp. 1950036 ◽

Cited By ~ 1

Author(s):

Noé Cuneo ◽

Vojkan Jakšić ◽

Claude-Alain Pillet ◽

Armen Shirikyan

Keyword(s):

Multifractal Analysis ◽

Invariant Measures ◽

Large Deviation ◽

Thermodynamic Formalism ◽

Hard Core ◽

Shift Spaces ◽

Large Deviation Principles ◽

Fluctuation Relation ◽

Level 3 ◽

Level 1

We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle–Lanford functions, and the exposition is essentially self-contained.

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