On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold

TL;DR

This paper demonstrates that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, but such behavior is rare, with typical growth being subexponential, addressing Arnold's conjecture.

Contribution

It provides explicit examples of rapid growth and establishes that such growth is exceptional, contributing to the understanding of intersection multiplicities in holomorphic dynamics.

Findings

Explicit examples of arbitrarily fast growth of intersection multiplicities

Most cases exhibit subexponential growth of intersection multiplicities

Addresses and partially resolves Arnold's conjecture on intersection growth

Abstract

We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.