Dynamical properties of simplicial systems and continued fraction algorithms

Abstract

In the first half of this study, we introduce a notion of simplicial systems that generalize the Rauzy graphs of interval exchange maps. We then show an effective criterion on them which imply many dynamical properties of Rauzy-Veech induction in this broader setting.In the second half, we show, on a large set of examples, that this formalism contains many classical multidimensional continued fraction algorithms. As a consequence, we obtain ergodicity as well as existence and uniqueness of a measure of maximal entropy on a canonical suspension of Brun, Selmer algorithms in all dimensions and of Arnoux-Rauzy-Poincaré in dimension three.In parallel, we consider fractal sets described in this formalism and show an explicit upper bound on their Hausdorff dimension as well as a construction of a measure of maximal entropy equivalent to its Hausdorff measure. This implies in particular that the Rauzy gasket in all dimensions has Hausdorff dimension strictly smaller than its ambient space.

An upper bound on the dimension of the Rauzy gasket

“…Empirical estimates in [5] suggest dim H (G) ≈ 1.72, and a lower bound was shown in [11]. Lastly, Fougeron used semiflows and thermodynamic techniques to show dim H (G) < 1.825 [9]. Using completely elementary methods, we show the following improved upper bound.…”

“…Remark 4.2. Perhaps surprisingly, this lemma appears to give a significant improvement on working exclusively with either diam(∆ i ) or area(∆ i ) (as in [9] and [1]).…”

An upper bound on the dimension of the Rauzy gasket

“…Empirical estimates in [4] suggest dim H (G) ≈ 1.72, and a lower bound was shown in [10]. Lastly, Fougeron used semiflows and thermodynamic techniques to show dim H (G) < 1.825 [8]. Using completely elementary methods, we show the following improved upper bound.…”

“…Remark 4.2. Perhaps surprisingly, this lemma appears to give a significant improvement on working exclusively with either diam(∆ i ) or area(∆ i ) (as in [8] and [1]).…”

Arnoux-Rauzy interval exchanges

“…In all approaches, the Rauzy gasket plays the role of parameters space. This prompted several authors to make deep studies of this gasket in [6] [8] [9] [27] [25], solving in this special case a conjecture of Novikov, and to look at everything that can be found about this particular family; it was also considered in Lemma 5.9 (attributed to Yoccoz) of [30], a paper related to geometric group theory and the study of automorphisms of free groups. But indeed, a priori not much is known, as these six-interval exchange transformations are only semi-conjugate to the original Arnoux-Rauzy systems.…”