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On the (Dis)connection Between Growth and Primitive Periodic Points

Adi Glücksam, Shira Tanny

TL;DR

The paper proves that growth constraints for holomorphic maps in several complex variables do not constrain the honest count of primitive periodic points: for any prescribed sequence of periods and point counts, there exists a holomorphic map $F:\mathbb{C}^2\to\mathbb{C}^2$ with order-zero growth $\log M_F(r)=O((\log r)^2)$ that exhibits at least the given numbers of $p$-primitive periodic points within expanding balls. The authors achieve this by combining a dispatcher construction—enabled by Hörmander's $\bar{\partial}$-theorem and carefully designed subharmonic functions—to stitch region-specific dynamics into a global map, and by modifying the Cornalba–Shiffman framework to produce isolated, period-rigid points across multiple prescribed periods. The result extends Cornalba–Shiffman’s two-variable counterexample to a dynamic setting, showing that the coarse zero count can be bounded by growth while the honest count can grow arbitrarily fast. This work provides a constructive bridge between growth control and complex dynamical behavior in higher dimensions, with potential implications for transcendental dynamics and zero-set theory.

Abstract

In 1972, Cornalba and Shiffman showed that the number of zeros of an order zero holomorphic function in two or more variables can grow arbitrarily fast. We generalize this finding to the setting of complex dynamics, establishing that the number of isolated primitive periodic points of an order zero holomorphic function in two or more variables can grow arbitrarily fast as well. This answers a recent question posed by L. Buhovsky, I. Polterovich, L. Polterovich, E. Shelukhin and V. Stojisavljević.

On the (Dis)connection Between Growth and Primitive Periodic Points

TL;DR

The paper proves that growth constraints for holomorphic maps in several complex variables do not constrain the honest count of primitive periodic points: for any prescribed sequence of periods and point counts, there exists a holomorphic map with order-zero growth that exhibits at least the given numbers of -primitive periodic points within expanding balls. The authors achieve this by combining a dispatcher construction—enabled by Hörmander's -theorem and carefully designed subharmonic functions—to stitch region-specific dynamics into a global map, and by modifying the Cornalba–Shiffman framework to produce isolated, period-rigid points across multiple prescribed periods. The result extends Cornalba–Shiffman’s two-variable counterexample to a dynamic setting, showing that the coarse zero count can be bounded by growth while the honest count can grow arbitrarily fast. This work provides a constructive bridge between growth control and complex dynamical behavior in higher dimensions, with potential implications for transcendental dynamics and zero-set theory.

Abstract

In 1972, Cornalba and Shiffman showed that the number of zeros of an order zero holomorphic function in two or more variables can grow arbitrarily fast. We generalize this finding to the setting of complex dynamics, establishing that the number of isolated primitive periodic points of an order zero holomorphic function in two or more variables can grow arbitrarily fast as well. This answers a recent question posed by L. Buhovsky, I. Polterovich, L. Polterovich, E. Shelukhin and V. Stojisavljević.

Paper Structure

This paper contains 9 sections, 5 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary Results on Subharmonic functions
  3. The proof
  4. Constructing the "dispatcher" functions
  5. Step 1: Constructing an appropriate subharmonic function.
  6. Step 2: The model map and the holomorphic approximation.
  7. Step 3: Bounding the error.
  8. Step 4: Bounding the growth.
  9. Modifying Cornalba-Shiffman's construction

Key Result

Theorem 1.3

For every sequence of periods, $\left\{{p_n}\right\}_{n=1}^\infty$, and for every rate, $\left\{{m_n}\right\}_{n=1}^\infty$, there exists a holomorphic function $F:\mathbb C^2\rightarrow\mathbb C^2$ satisfying

Figures (2)

  • Figure 1: An illustration of the the image of the subharmonic function created by Lemma \ref{['lem:punctures']}
  • Figure 2: An illustration of the zeroes of a symmetrized Cornalba--Shiffman function, for periods $\{p_n\}=\{4,8,4,...\}$ and rates $\{m_n\}=\{2,2,4,...\}$.

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.3
  • Lemma 2.1: The Puncture Lemma
  • proof
  • Claim 2.2: evdoridou2023unbounded
  • Lemma 3.1
  • Remark 3.2
  • proof
  • Theorem 3.3
  • Example 3.4
  • ...and 4 more