Matching for a family of infinite measure continued fraction transformations

TL;DR

This paper investigates a family of infinite measure continued fraction transformations, establishing matching properties for most parameters, and derives explicit formulas for their invariant measures and entropy, advancing understanding of these dynamical systems.

Contribution

It introduces a new one-parameter family of continued fraction maps with an indifferent fixed point, proving matching for almost all parameters and constructing their natural extensions.

Findings

Matching holds for Lebesgue almost every parameter.

Explicit density of the invariant measure is derived.

Krengel entropy and other dynamical quantities are computed.

Abstract

As a natural counterpart to Nakada's $\alpha$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $\sigma$-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.