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Equidistribution of graphs of holomorphic correspondences

Muhan Luo

TL;DR

This paper proves exponential equidistribution of graphs of iterates for holomorphic correspondences on a compact Riemann surface, distinguishing regimes with distinct and equal dynamical degrees. In the $d_1<d_2$ case, an equilibrium measure $\mu$ satisfies $f^*(\mu)=d_2\mu$ and the normalized graph currents converge to $\Gamma_\infty=\pi_1^*(\mu)$ at an explicit exponential rate for $C^\alpha$ test forms. In the equal-degree, non-weakly modular case, invariant measures $\mu^+$ and $\mu^-$ yield $\Gamma_\infty=\pi_1^*(\mu^+)+\pi_2^*(\mu^-)$ with a similar exponential convergence. The authors deploy interpolation (reducing to $C^5$ tests), local chart analysis, Fourier expansions, and equidistribution properties of functions to obtain rate-controlled convergence, and extend the results to cycles with positive real coefficients and to certain higher-dimensional meromorphic settings.

Abstract

Let $X$ be a compact Riemann surface. Let $f$ be a holomorphic self-correspondence of $X$ with dynamical degrees $d_1$ and $d_2$. Assume that $d_1\neq d_2$ or $f$ is non-weakly modular. We show that the graphs of the iterates $f^n$ of $f$ are equidistributed exponentially fast with respect to a positive closed current in $X\times X$.

Equidistribution of graphs of holomorphic correspondences

TL;DR

This paper proves exponential equidistribution of graphs of iterates for holomorphic correspondences on a compact Riemann surface, distinguishing regimes with distinct and equal dynamical degrees. In the case, an equilibrium measure satisfies and the normalized graph currents converge to at an explicit exponential rate for test forms. In the equal-degree, non-weakly modular case, invariant measures and yield with a similar exponential convergence. The authors deploy interpolation (reducing to tests), local chart analysis, Fourier expansions, and equidistribution properties of functions to obtain rate-controlled convergence, and extend the results to cycles with positive real coefficients and to certain higher-dimensional meromorphic settings.

Abstract

Let be a compact Riemann surface. Let be a holomorphic self-correspondence of with dynamical degrees and . Assume that or is non-weakly modular. We show that the graphs of the iterates of are equidistributed exponentially fast with respect to a positive closed current in .
Paper Structure (3 sections, 6 theorems, 48 equations)

This paper contains 3 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction and Main result
  2. Preliminary Results
  3. Proof of the main theorems

Key Result

Theorem 1.1

Let $f$ be a holomorphic correspondence on a compact Riemann surface $X$ with degrees $d_1<d_2$. Let $\mu,\Gamma_n$ and $\Gamma_\infty$ be as above. Then for every $\alpha>0$, there is a constant $0<\lambda_\alpha<1$ such that for any test $(1,1)$-form $\beta$ of class $\mathcal{C}^\alpha$ on $X\tim where $C_\alpha>0$ is a constant independent of $n$ and $\beta$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: DKW, Definition 3.1
  • Proposition 2.2: DKW, Proposition 3.1
  • Proposition 2.3: DKW, Proposition 3.2
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • proof : End of the proof of Theorem \ref{['thm:main-thm-neq']}
  • ...and 2 more