[Paper Review] Synchronization is full measure for all $\alpha$-deformations of an infinite class of continued fraction transformations

Kariane Calta, Cor Kraaikamp+1 more|arXiv (Cornell University)|Jan 17, 2017

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes that for an infinite family of α-deformed continued fraction maps associated with triangle Fuchsian groups G3,n, the set of parameters α for which the orbits of the interval endpoints synchronize has full Lebesgue measure. Using a tree of admissible words and group-theoretic identities, the authors fully characterize the synchronization sets across three parameter regimes, proving that non-synchronization occurs only on a measure-zero Cantor set, thereby enabling the construction of natural extensions and entropy functions for these maps.

ABSTRACT

We study an infinite family of one-parameter deformations, so-called $\alpha$-continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the entropy of the map indexed by $\alpha$ varies in a way directly related to whether or not the orbits of the endpoints of the map synchronize. For two cases of one-parameter deformations associated to the classical case of the modular group $ ext{PSL}_2(\mathbb Z)$, the set of $\alpha$ for which synchronization occurs has been determined. Here, we explicitly determine the synchronization sets for each $\alpha$-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $\alpha$-expansions back to regular continued fraction expansions is not available to us.) A curiosity here is that all of our synchronization sets can be described in terms of a single tree of words. In a paper in preparation, we identify the natural extensions of our maps, as well as the entropy functions associated to each deformation.

Motivation & Objective

• To fully characterize the set of α-values for which the orbits of the endpoints of α-deformed continued fraction maps synchronize.
• To extend the theory of synchronization beyond the modular group to an infinite family of triangle Fuchsian groups G3,n.
• To provide a complete description of synchronization sets using a single tree of admissible words, even when regular continued fraction expansions are not available.
• To lay the foundation for determining natural extensions and entropy functions of these maps in a forthcoming paper.

Proposed method

• Introduces a one-parameter family of interval maps T3,n,α associated with triangle Fuchsian groups G3,n, generalizing classical α-continued fractions.
• Uses a tree of admissible words V to index synchronization intervals, with admissibility determined by digit expansions and full-branched properties of word sequences.
• Applies group element identities involving matrices A, B, C, and U to relate orbit dynamics to algebraic relations in G3,n.
• Employs induction and ergodicity arguments (from the α=0 case) to prove that the complement of synchronization intervals is a Cantor set of measure zero.
• Analyzes three parameter regimes: α < γ3,n, α > ϵ3,n, and γ3,n < α < ϵ3,n, each requiring distinct word-based and group-theoretic arguments.
• Establishes synchronization via the condition that both left and right orbit expansions remain admissible, with endpoints determined by loss of admissibility.

Experimental results

Research questions

• RQ1 For which α-values do the orbits of the left and right endpoints of the interval map T3,n,α eventually synchronize?
• RQ2 Can the synchronization set for α-deformations of non-modular triangle Fuchsian groups be fully described without relying on regular continued fraction expansions?
• RQ3 How do group-theoretic identities in G3,n relate to the structure of synchronization intervals in the parameter space?
• RQ4 Is the set of non-synchronizing α-values negligible in measure for these deformations?
• RQ5 Can the synchronization structure be encoded uniformly via a single tree of words across all n ≥ 3?

Key findings

• For all n ≥ 3, the set of α ∈ (0,1) for which the T3,n,α-orbits of the left and right endpoints synchronize has full Lebesgue measure.
• The synchronization set is a disjoint union of intervals indexed by Z≠0 × V, where V is a tree of admissible words, with each interval corresponding to a pair of admissible digit expansions.
• The complement of the synchronization set is a Cantor set of Lebesgue measure zero, proven using the ergodicity of the α=0 map.
• Synchronization occurs precisely when both the left and right orbit expansions remain admissible, with endpoints determined by the first loss of admissibility.
• The synchronization intervals are fully characterized by group identities involving matrices A, C, and U, with explicit formulas for the endpoints of each interval.
• The structure of the synchronization set is uniform across all n ≥ 3, with the tree of words V providing a universal encoding mechanism independent of n.