Large deviations for some unbounded observables in dynamical systems

In this paper we establish a large deviations type estimate for strongly mixing Markov chains with respect to the Lp norm. As applications we derive such estimates for the iterates of a locally constant random cocycle with mixed rank, as well as for unbounded observables of expanding maps.

💡 Research Summary

The paper addresses a gap in the large‑deviation theory for dynamical systems and Markov chains when the observable is unbounded. Classical large‑deviation principles (LDP) are well‑understood for bounded observables belonging to a Banach space with a spectral gap (e.g., Hölder or bounded‑variation functions). However, many natural observables—such as logarithmic distance to a singular point, Lyapunov exponents, or entropy production—are unbounded, and the usual spectral‑gap techniques cannot be applied directly.

The authors develop a systematic framework that extends exponential‑type deviation estimates to a broad class of unbounded observables. The key idea is to split the observable φ into a truncated bounded part φ_M = φ·1_{φ≤M} and a tail part R_M = φ·1_{φ>M}. The bounded part belongs to the chosen Banach space B (for instance, the Hölder space C^α) and therefore inherits the strong mixing properties of the underlying Markov system. The tail part is handled by imposing two quantitative conditions: (i) an exponential tail bound μ(φ>t) ≤ C₁ e^{−αt} and (ii) an L²‑control of the propagated tail, \