Additive, almost additive and asymptotically additive potential sequences are equivalent

TL;DR

This paper proves that almost additive and asymptotically additive potential sequences in dynamical systems are essentially equivalent to standard additive potentials, simplifying the theoretical framework and unifying various results.

Contribution

The paper demonstrates that all such potential sequences are equivalent to standard potentials, showing they do not extend the theory beyond classical additive potentials.

Findings

Almost additive and asymptotically additive sequences are equivalent to standard potentials.

Many results for these sequences follow directly from classical additive potential theory.

All quasi-Bernoulli measures are shown to be weak Gibbs.

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.