Time-Changes of Heisenberg nilflows

TL;DR

This paper studies the dynamics of Heisenberg nilflows, constructing special functionals with scaling properties, and proves that generic time-changes lead to mixing with polynomial or stretched polynomial decay of correlations.

Contribution

It introduces Bufetov functionals for Heisenberg nilflows, proving their analyticity and using this to establish mixing and decay of correlations for generic time-changes.

Findings

Generic non-trivial time-changes are mixing.

Decay of correlations can be polynomial or stretched polynomial.

Bufetov functionals exhibit modularity and scaling properties.

Abstract

We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions, have a modular property and scale exactly under the renor- malization dynamics. We then prove analyticity of the functionals in the transverse directions to the flow. As a consequence of this analyticity property we derive that there exists a full measure set of nilflows such that generic (non-trivial) time-changes are mixing and moreover have a "stretched polynomial" decay of correlations for sufficiently smooth functions. Moreover we also prove that there exists a full Hausdorff dimension set of nilflows such that generic non-trivial time-changes have polynomial decay of correlations.