# Dynamics of a family of continued fraction maps

## TL;DR

This paper investigates a family of continued fraction maps, including the Gauss and Fibonacci maps, analyzing their transfer operators, invariant measures, and a related involution, revealing common functional equations and specific invariant measures.

## Contribution

It introduces a unified framework for analyzing a family of continued fraction maps, deriving their transfer operators and invariant measures, and exploring their interrelations through an involution.

## Key findings

- Invariant measures satisfy a common functional equation.
- Explicit invariant measures found for some family members.
- An involution relates the Gauss and Fibonacci maps.

## Abstract

We study the dynamics of a family of continued fraction maps parametrized by the unit interval. This family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these dynamical maps and make a preliminary study of them. We show that their analytic invariant measures obeys a common functional equation generalizing Lewis' functional equation and we find invariant measures for some members of the family. We also discuss a certain involution of this family which sends the Gauss map to the Fibonacci map.