Bundle switching in higher dimensions

TL;DR

This paper explores how certain three-dimensional dynamical systems can be extended to higher dimensions, resulting in new examples of partially hyperbolic diffeomorphisms that are dynamically incoherent, with implications for understanding complex dynamical behaviors.

Contribution

It introduces a method to construct higher-dimensional partially hyperbolic diffeomorphisms from three-dimensional systems, demonstrating the occurrence of dynamical incoherence in these higher-dimensional cases.

Findings

Construction of 4D dynamically incoherent partially hyperbolic diffeomorphisms from 3D systems

Extension of results to higher dimensions under additional assumptions

Identification of conditions preserving orientation of stable bundles

Abstract

Assuming it preserves an orientation of its stable bundle, any three-dimensional partially hyperbolic diffeomorphism can be used to construct a four-dimensional partially hyperbolic diffeomorphism which is dynamically incoherent. Under the same assumption, the time-one map of any three-dimensional Anosov flow can be used to construct a four-dimensional diffeomorphism which is both absolutely partially hyperbolic and dynamically incoherent. Further results hold in higher dimensions under additional assumptions.