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On holomorphic partially hyperbolic systems

Disheng Xu, Jiesong Zhang

Abstract

We construct examples illustrating that dynamically-defined distributions of holomorphic diffeomorphisms on compact complex manifolds are not necessarily holomorphic in any open subset. More precisely, for any $n\geq 5$, we construct a holomorphic fibered partially hyperbolic system on a complex $n$-fold, where the center distribution is not holomorphic in any open subset. For $n=3$ we demonstrate a contrast: the center distribution of any fibered holomorphic partially hyperbolic diffeomorphism on a complex $3$-fold is holomorphic. In particular, any such a system is a holomorphic skew product over a linear automorphism on a complex $2$-torus.

On holomorphic partially hyperbolic systems

Abstract

We construct examples illustrating that dynamically-defined distributions of holomorphic diffeomorphisms on compact complex manifolds are not necessarily holomorphic in any open subset. More precisely, for any , we construct a holomorphic fibered partially hyperbolic system on a complex -fold, where the center distribution is not holomorphic in any open subset. For we demonstrate a contrast: the center distribution of any fibered holomorphic partially hyperbolic diffeomorphism on a complex -fold is holomorphic. In particular, any such a system is a holomorphic skew product over a linear automorphism on a complex -torus.
Paper Structure (32 sections, 56 theorems, 86 equations)

This paper contains 32 sections, 56 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. A brief introduction to Anosov systems, partially hyperbolic systems and pathological foliations
  3. Holomorphic diffeomorphisms and holomorphic rigidity
  4. Non-holomorphic dynamically-defined invariant distributions
  5. Low dimensional holomorphic rigidity and results on accessibility classes
  6. Sketch of the proof and plan of the article
  7. Uniformly quasiconformal partially hyperbolic systems, heat flows, the existence of Gibbs $cu$-states and $C^\infty$ property of $E^c$.
  8. Deformation of complex structures (I): moduli space of $\mathbb T^2$ and holomorphicity of $E^c$ for $3$d case.
  9. Deformation of complex structures (II): Non-Kähler complex structures on higher dimensional tori and constructions of non-holomorphic center distributions.
  10. Further problems
  11. Preliminaries
  12. Preliminaries in complex geometry
  13. Topological facts about fibered partially hyperbolic system
  14. Bunching and holomophy of the stable and unstable holonomies
  15. Disintegration of measures and Gibbs measures
  16. ...and 17 more sections

Key Result

Theorem 1.2

For any $n \geqslant 5$, there exists a compact complex manifold $M^n$ of dimension $n$ and a fibered holomorphic partially hyperbolic diffeomorphism $f$ on $M^n$ such that the center distribution is real analytic but not holomorphic on any non-empty open subset (implying that the center fibration $

Theorems & Definitions (96)

  • Definition 1.1
  • Conjecture 1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Lemma 2.1
  • ...and 86 more