# Kneading sequences for toy models of H\'enon maps

## TL;DR

This paper introduces kneading sequences for a two-dimensional toy model of Hénon maps, demonstrating they serve as complete invariants for conjugacy classes and establishing a version of Singer's theorem.

## Contribution

It extends the concept of kneading sequences to a 2D setting and proves their completeness as invariants for the toy model family of Hénon maps.

## Key findings

- Kneading sequences are complete invariants for conjugacy classes.
- A version of Singer's theorem is established for the toy model.
- The study links combinatorial equivalence with topological conjugacy in 2D dynamics.

## Abstract

The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of H\'enon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer's Theorem for the toy model family.