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A view towards mixing in holomorphic correspondences

Sathi Trikkadeeri Mana, Bharath Krishna Seshadri

Abstract

In this manuscript we develop a theory of mixing and weakly mixing in the study of dynamics of holomorphic correspondences defined on a compact connected complex manifold. We also connect these notions to the theory of ergodicity of holomorphic correspondences developed by Londhe. Further, we give motivation and illustrative examples that compare the present scenario with that of maps. Finally, we study product of two holomorphic correspondences and use them to characterise weakly mixing.

A view towards mixing in holomorphic correspondences

Abstract

In this manuscript we develop a theory of mixing and weakly mixing in the study of dynamics of holomorphic correspondences defined on a compact connected complex manifold. We also connect these notions to the theory of ergodicity of holomorphic correspondences developed by Londhe. Further, we give motivation and illustrative examples that compare the present scenario with that of maps. Finally, we study product of two holomorphic correspondences and use them to characterise weakly mixing.

Paper Structure

This paper contains 22 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. An invitation to holomorphic correspondences
  3. The dynamics
  4. Measure theoretic aspects
  5. The Koopman operator
  6. Proof:
  7. Mixing definition and motivation
  8. Proof:
  9. Ergodicity through the lens of mixing
  10. Proof:
  11. Proof:
  12. Proof:
  13. Proof:
  14. Proof:
  15. Proof:
  16. ...and 7 more sections

Key Result

Theorem 2.3

Let $F$ be a holomorphic correspondence on a compact Kähler manifold $X$ of dimension $k$ with $d > d_{k - 1}$, where $d_{k - 1}$ is the dynamical degree of $F$ of order $k - 1$ and $d$ is the topological degree of $F$. Then, there is a pluripolar set $\mathcal{E} \subsetneq X$ such that Moreover, the measure $\mu_F$ is $F^{*}$ invariant and $\mu_F(\mathcal{E}) = 0$. Here, $\delta_{z}$ denotes th

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Dinh - Sibony, dinhsibony_og
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Londhe, londhe_ergodicity
  • Theorem 2.7: Londhe, londhe_ergodicity
  • Definition 2.8
  • Proposition 2.9
  • Definition 3.1
  • ...and 16 more