Structure of attractors for (a,b)-continued fraction transformations

TL;DR

This paper investigates the attractor structures of a two-parameter family of (a,b)-continued fraction maps, revealing finite rectangular attractors for most parameters and describing the exceptional set, with implications for invariant measures and ergodic theory.

Contribution

It characterizes the attractor structures of (a,b)-continued fractions, proving finite rectangular attractors for almost all parameters and describing the exceptional set, advancing understanding of their ergodic properties.

Findings

Attractors have finite rectangular structure for almost all parameters.

The exceptional parameter set is a Cantor-like set of measure zero.

The theory applies to invariant measures and ergodic properties of related maps.

Abstract

We study a two-parameter family of one-dimensional maps and related (a,b)-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed" from the data (a,b), and that for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.