# Hyperbolicity of renormalization for dissipative gap mappings

## TL;DR

This paper proves the hyperbolicity of renormalization for dissipative gap mappings, a class of one-dimensional dynamical systems with applications to higher-dimensional flows, and characterizes their conjugacy classes as smooth manifolds.

## Contribution

It establishes hyperbolicity of renormalization for $C^3$ dissipative gap mappings and describes the structure of their conjugacy classes as $C^1$ manifolds.

## Key findings

- Hyperbolicity of renormalization for dissipative gap mappings.
- Topological conjugacy classes form $C^1$ manifolds.
- Application to understanding higher-dimensional flow dynamics.

## Abstract

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.