Additive, almost additive and asymptotically additive potential sequences are equivalent

Abstract

Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually equivalent to a standard (additive) potential in the sense that there exists a continuous potential with the same topological pressure, equilibrium states, variational principle, weak Gibbs measures, level sets (and irregular set) for the Lyapunov exponent and large deviations properties. In this sense, our result shows that almost and asymptotically additive potential sequences do not extend the scope of the theory compared to standard potentials, and that many results in the literature about such sequences can be recovered as immediate consequences of their counterpart in the additive case. A corollary of our main result is that all quasi-Bernoulli measures are weak Gibbs.

References in corpus (10)

• Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
• Lyapunov spectrum of asymptotically sub-additive potentials
• On factors of Gibbs measures for almost additive potentials
• Multifractal analysis of the irregular set for almost-additive sequences via large deviations
• Almost-additive thermodynamic formalism for countable Markov shifts
• Large Deviations and Fluctuation Theorem for Selectively Decoupled Measures on Shift Spaces
• Strengthened large deviations for rational maps and full shifts, with unified proof
• Zero temperature limits of Gibbs states for almost-additive potentials
• Weak Specification Properties and Large Deviations for Non-additive Potentials
• Weak Gibbs measures and large deviations

Cited by in corpus (5)

• On entropy production of repeated quantum measurements II. Examples
• Large deviations of return times and related entropy estimators on shift spaces
• Nonadditive families of potentials: physical equivalence and some regularity relations
• Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows
• On Gibbs measures for almost additive sequences associated to some relative pressure functions