# Negative Lyapunov exponent of circle maps forced by expanding circle endomorphisms

## TL;DR

This paper investigates certain torus maps with expanding base dynamics and non-monotonic fiber maps, demonstrating that under typical conditions, the fiber Lyapunov exponents are negative almost everywhere, leading to local synchronization of orbits.

## Contribution

The paper identifies an open class of $C^1$-maps on the torus with negative fiber Lyapunov exponents and establishes uniform bounds depending on fiber characteristics.

## Key findings

- Lyapunov exponents on fibers are negative almost everywhere.
- Uniform upper bounds for Lyapunov exponents depend on fiber map properties.
- Orbits on the same fiber tend to synchronize locally.

## Abstract

We study maps on the torus $\mathbb{T}^2$ that are of the form $F(x,y) = (bx, f_x(y))$, where $b\geq 2$ is an integer. We establish an open class of $C^1$-maps, with $f_x(y)$ that are typically non-monotonic in $x$, for which the Lyapunov exponents on the fibre are negative almost everywhere. For each fixed $f_x(y)$ and a base map $bx$ that is sufficiently expanding, we establish a uniform upper bound for the Lyapunov exponents; moreover, the uniform bound depends on selective characteristics of $f$. This implies that orbits on the same fibre exhibit local synchronisation.