On some symmetric multidimensional continued fraction algorithms

Abstract

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear that it is of positive measure.

Version 1: 22 pages, 12 figures. Version 2: new section on Cassaigne algorithm, 25 pages, 15 figures. Version 3: corrections during review process

References in corpus (2)

• Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincaré Algorithm
• Natural extensions and Gauss measures for piecewise homographic continued fractions

Cited by in corpus (8)

• A Set of Sequences of Complexity $2n+1$
• Almost everywhere balanced sequences of complexity $2n+1$
• $3$-dimensional Continued Fraction Algorithms Cheat Sheets
• Natural extensions and Gauss measures for piecewise homographic continued fractions
• $\mathbf{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometry
• Symbolic coding of linear complexity for generic translations of the torus, using continued fractions
• On The Dimension Group of Unimodular S-Adic Subshifts
• On Cross Sections to the Geodesic and Horocycle Flows on Quotients of $\mathrm{SL}(2,\mathbb{R})$ by Hecke Triangle Groups $G_q$