Surgery for partially hyperbolic dynamical systems II. Blow-up of a complex curve

TL;DR

This paper introduces a novel slow-down construction combined with blow-up surgery to create new higher-dimensional partially hyperbolic flows, expanding the class of known volume-preserving and non-fiberwise Anosov examples.

Contribution

It develops a new slow-down technique that relaxes previous domination assumptions, enabling the construction of diverse partially hyperbolic flows, including volume-preserving and non-fiberwise Anosov types.

Findings

Constructed new volume-preserving partially hyperbolic flows.

Produced examples not fiberwise Anosov.

Relaxed domination assumptions in hyperbolic flow construction.

Abstract

In this paper we use the blow-up surgery introduced in [G] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [G]. Consequently we gain more flexibility which allows to construct new volume-preserving partially hyperbolic flows as well as new examples which are not fiberwise Anosov. The latter are produced by starting with the geodesic flow on complex hyperbolic manifold which admits a totally geodesic complex curve. Then by performing the slow-down first and the blow-up second we obtain a new (volume-preserving) partially hyperbolic flows.