On sensitive dependence on initial conditions and existence of physical measure for 3-flows

TL;DR

This paper demonstrates that in three-dimensional flows, robust chaos characterized by sensitivity to initial conditions is equivalent to the presence of hyperbolic structures and physical measures, linking chaotic behavior to measurable statistical properties.

Contribution

It establishes the equivalence between robust chaos, hyperbolic structures, and physical measures in three-dimensional flows, clarifying the conditions for chaos to be statistically observable.

Findings

Robust chaotic behavior implies the existence of hyperbolic structures.

Hyperbolic structures are associated with physical measures.

In low dimensions, chaos ensures the existence of a physical measure.

Abstract

After reviewing known results on sensitiveness and also on robustness of attractors together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic behavior meaning sensitiveness to initial conditions for the past as well for the future for all nearby flows) is equivalent to the existence of certain hyperbolic structures. These structures, in turn, are associated to the existence of physical measures. In short in low dimensions, robust chaotic behavior for smooth flows ensures the existence of a physical measure.