Observable Optimal State Points of Sub-additive Potentials

TL;DR

This paper extends Dai's method for selecting points with negative growth rates in sub-additive potentials from ergodic systems to non-ergodic systems, ensuring such points exist in a positive measure set under mild conditions.

Contribution

It generalizes Dai's results to non-ergodic systems, showing the existence of negative growth points in a positive measure set without requiring measure preservation.

Findings

Negative growth points exist in non-ergodic systems.

Such points can be found in a positive Lebesgue measure set.

The results do not require the system to preserve an absolutely continuous measure.

Abstract

For a sequence of sub-additive potentials, Dai [Optimal state points of the sub-additive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573] gave a method of choosing state points with negative growth rates for an ergodic dynamical system. This paper generalizes Dai's result to the non-ergodic case, and proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.