Large deviation principle in one-dimensional dynamics

TL;DR

This paper proves a comprehensive large deviation principle for certain one-dimensional interval maps, including some without physical measures, challenging traditional views on large deviations in dynamical systems.

Contribution

It establishes the full level-2 Large Deviation Principle for topologically exact smooth interval maps with non-flat critical points, including non-renormalizable quadratic maps.

Findings

Large deviation principle holds for all such maps

Includes maps without physical measures

Challenges the view of large deviations as just a refinement of laws of large numbers

Abstract

We study the dynamics of smooth interval maps with non-flat critical points. For every such a map that is topologically exact, we establish the full (level-2) Large Deviation Principle for empirical means. In particular, the Large Deviation Principle holds for every non\nobreakdash-renormalizable quadratic map. This includes the maps without physical measure found by Hofbauer and Keller, and challenges the widely-shared view of the Large Deviation Principle as a refinement of laws of large numbers.