Natural extensions and Gauss measures for piecewise homographic continued fractions

Arnoux, Pierre; Schmidt, Thomas A.

doi:10.24033/bsmf.2791

Natural extensions and Gauss measures for piecewise homographic continued fractions
[Extensions naturelles et mesures de Gauss pour des algorithmes de fraction continues homographiques par morceaux]

Arnoux, Pierre ¹ ; Schmidt, Thomas A.²

¹ Université d’Aix-Marseille et CNRS, Institut de Mathématiques de Marseille (UMR7373), Site Sud, Campus de Luminy, Case 907, 13288 MARSEILLE Cedex 9 France
² Department of Mathematics, Oregon State University Corvallis, OR 97331

Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 3, pp. 515-544

Abstract (VO)
Résumé

We give a heuristic method to solve explicitly for an absolutely continuous invariant measure for a piecewise differentiable, expanding map of a compact subset $I$ of Euclidean space $ℝ^{d}$ . The method consists of constructing a skew product family of maps on $I \times ℝ^{d}$ , which has an attractor. Lebesgue measure is invariant for the skew product family restricted to this attractor. Under reasonable measure-theoretic conditions, integration over the fibers gives the desired measure on $I$ . Furthermore, the attractor system is then the natural extension of the original map with this measure. We illustrate this method by relating it to various results in the literature.

Nous donnons une méthode heuristique pour trouver explicitement une mesure invariante absolument continue pour une application différentiable par morceaux et expansive d’un sous-ensemble compact $I$ de l’espace euclidien $ℝ^{d}$ . La méthode consiste à construire une famille d’applications qui est un produit croisé sur $I \times ℝ^{d}$ , et à montrer que cette famille possède un attracteur. La mesure de Lebesgue est par construction invariante pour la restriction à l’attracteur de cette famille d’applications. Sous des hypothèses raisonnables sur la mesure, l’intégration le long des fibres donne la mesure invariante cherchée sur $I$ . De plus, le système construit sur l’attracteur est l’extension naturelle de l’application originale, munie de cette mesure invariante. Nous illustrons cette méthode par plusieurs exemples tirés de la littérature.

MR

DOI : 10.24033/bsmf.2791

Classification : 37E05, 11K50, 37A45
Keywords: natural extensions, continued fractions, invariant measures, iterated function system
Mots-clés : extension naturelle, fractions continues, mesures invariantes, systèmes de fonctions itérées

Affiliations des auteurs :

Arnoux, Pierre ¹ ; Schmidt, Thomas A. ²

¹ Université d’Aix-Marseille et CNRS, Institut de Mathématiques de Marseille (UMR7373), Site Sud, Campus de Luminy, Case 907, 13288 MARSEILLE Cedex 9 France
² Department of Mathematics, Oregon State University Corvallis, OR 97331

@article{BSMF_2019__147_3_515_0,
     author = {Arnoux, Pierre and Schmidt, Thomas A.},
     title = {Natural extensions and {Gauss} measures for piecewise homographic continued fractions},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {515--544},
     year = {2019},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {147},
     number = {3},
     doi = {10.24033/bsmf.2791},
     mrnumber = {4030549},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2791/}
}

TY  - JOUR
AU  - Arnoux, Pierre
AU  - Schmidt, Thomas A.
TI  - Natural extensions and Gauss measures for piecewise homographic continued fractions
JO  - Bulletin de la Société Mathématique de France
PY  - 2019
SP  - 515
EP  - 544
VL  - 147
IS  - 3
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2791/
DO  - 10.24033/bsmf.2791
LA  - en
ID  - BSMF_2019__147_3_515_0
ER  -

%0 Journal Article
%A Arnoux, Pierre
%A Schmidt, Thomas A.
%T Natural extensions and Gauss measures for piecewise homographic continued fractions
%J Bulletin de la Société Mathématique de France
%D 2019
%P 515-544
%V 147
%N 3
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2791/
%R 10.24033/bsmf.2791
%G en
%F BSMF_2019__147_3_515_0

Arnoux, Pierre; Schmidt, Thomas A. Natural extensions and Gauss measures for piecewise homographic continued fractions. Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 3, pp. 515-544. doi: 10.24033/bsmf.2791

Bibliographie
Cité par

Adler, R.; Flatto, L. Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.), Volume 25 (1991), pp. 229-334 | MR | Zbl | DOI

Arnoux, P.; Hubert, P. Fractions continues sur les surfaces de Veech, J. Anal. Math., Volume 81 (2000), pp. 35-64 | MR | Zbl | DOI

Arnoux, P.; Labbé, S. On some symmetric multidimensional continued fraction algorithms, Ergodic Theory and Dynamical Systems (2017), pp. 1-26 | Zbl | MR | DOI

Arnoux, P.; Mizutani, M.; Sellami, T. Random product of substitutions with the same incidence matrix, Theoret. Comput. Sci., Volume 543 (2014), pp. 68-78 | MR | Zbl | DOI

Arnoux, P.; Nogueira, A. Mesures de Gauss pour des algorithmes de fractions continues multidimensionnelles, Ann. Sci. École Norm. Sup. (4), Volume 26 (1993), pp. 645-664 | MR | Zbl | Numdam | DOI

Arnoux, P. Le codage du flot géodésique sur la surface modulaire. (French) [Coding of the geodesic flow on the modular surface], Enseign. Math. (2), Volume 40 (1994), pp. 29-48 | MR | Zbl

Artin, E. Ein mechanisches System mit quasi-ergodischen Bahnen, Abh. Math. Sem. Hamburg, Volume 1982 (1924), pp. 170-175 | MR | JFM | DOI

Arnoux, P.; Schmidt, T. A. Cross sections for geodesic flows and $α$ -continued fractions, Nonlinearity, Volume 26 (2013), pp. 711-726 | MR | Zbl | DOI

Arnoux, P.; Schmidt, T. A. Commensurable continued fractions, Discrete and Continuous Dynamical Systems – Series A (DCDS-A), Volume 34 (2014), pp. 4389-4418 | MR | Zbl | DOI

Barnley, M. F. Fractals everywhere, Second edition. Revised with the assistance of and with a foreword by Hawley Rising, III, Academic Press Professional, Boston, MA, 1993, 534 pages | MR | Zbl

Bosma, W.; Jager, H.; Wiedijk, F. Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math., Volume 45 (1983), pp. 281-299 | MR | Zbl | DOI

Burton, R.; Kraaikamp, C.; Schmidt, T.A. Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., Volume 352 (2000), pp. 1277-1298 | MR | Zbl | DOI

Berthé, V.; Labbé, S. Factor complexity of $S$ -adic words generated by the Arnoux-Rauzy-Poincaré algorithm, Adv. Appl. Math., Volume 63 (2015), pp. 90-130 | MR | Zbl | DOI

Chan, H. C. A Gauss-Kuzmin-Lévy theorem for a certain continued fraction, International Journal of Mathematics and Mathematical Sciences, Volume 2004 (2004), pp. 1067-1076 | MR | Zbl | DOI

Calta, K.; Kraaikamp, C.; Schmidt, T. A. Synchronization is full measure for all $α$ -deformations of an infinite class of continued fraction transformations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (to appear; 50 pp) | MR

Carminati, C.; Tiozzo, G. A canonical thickening of $ℚ$ and the entropy of $α$ -continued fraction transformations, Ergodic Theory Dynam. Systems, Volume 32 (2012), pp. 1249-1269 | MR | Zbl | DOI

Dajani, K.; Kraaikamp, C.; Steiner, W. Metrical theory for $α$ -Rosen fractions, J. Eur. Math. Soc. (JEMS), Volume 11 (2009), pp. 1259-1283 | MR | Zbl | DOI

Ei, H.; Ito, S.; Nakada, H.; Natsui, R. On the construction of the natural extension of the Hurwitz complex continued fraction map, Monatsh. Math., Volume 188 (2019), pp. 37-86 | MR | DOI

Gauss, C. F. Letter to Laplace, Gauss Werke, 1917, pp. 371-374

Gröchenig, K.; Haas, A. Backward continued fractions and their invariant measures, Canad. Math. Bull., Volume 39 (1996), pp. 186-198 | MR | Zbl | DOI

Hilgert, J.; Pohl, A. Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one, Infinite dimensional harmonic analysis IV, NJ, World Sci. Publ., Hackensack, 2009 | MR | Zbl

Hutchinson, J. Fractals and self similarity, Indiana Univ. Math. J., Volume 30 (1981), pp. 713-747 | MR | Zbl | DOI

Keane, M. A continued fraction titbit, Symposium in Honor of Benoit Mandelbrot (Curaçao, 1995), Fractals, Volume 3 (1995), pp. 641-650 | MR | Zbl

Kraaikamp, C. A new class of continued fraction expansions, Acta Arith., Volume 1 (1991), pp. 1-39 | MR | Zbl | DOI

Kraaikamp, C.; Schmidt, T. A.; Steiner, W. Natural extensions and entropy of $α$ -continued fractions, Nonlinearity, Volume 25 (2012), pp. 2207-2243 | MR | Zbl | DOI

Katok, S.; Ugarcovici, I. Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.), Volume 44 (2007), pp. 87-132 | MR | Zbl | DOI

Katok, S.; Ugarcovici, I. Structure of attractors for $(a, b)$ -continued fraction transformations, J. Mod. Dyn., Volume 4 (2010), pp. 637-691 | MR | Zbl | DOI

Luzzi, L.; Marmi, S. On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., Volume 20 (2008), pp. 673-711 | MR | Zbl | DOI

Moussa, P.; Cassa, A.; Marmi, S. Continued fractions and Brjuno functions, J. Comput. Appl. Math., Volume 105 (1999), pp. 403-415 | MR | Zbl | DOI

Mayer, D.; Mühlenbruch, T. Nearest $λ_{q}$ -multiple fractions, Spectrum and dynamics, Volume 52 (2010), pp. 147-184 | MR | Zbl | DOI

Mayer, D.; Strömberg, F. Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., Volume 2 (2008), pp. 581-627 | MR | Zbl | DOI

Nakada, H. Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., Volume 4 (1981), pp. 399-426 | MR | Zbl | DOI

Nakada, H. Continued fractions, geodesic flows and Ford circles, Algorithms, fractals, and dynamics: Okayama/Kyoto, Volume 1992 (1995), pp. 179-191 | MR | Zbl | DOI

Nakada, H.; Ito, S.; Tanaka, S. On the invariant measure for the transformations associated with some real continued-fractions, Keio Engrg. Rep., Volume 30 (1977), pp. 159-175 | MR | Zbl

Ralston, D. Substitutions and 1/2-discrepancy of ${n θ + x}$ , Acta Arith., Volume 154 (2012), pp. 1-28 | MR | Zbl | DOI

Ralston, D. 1/2 -Heavy sequences driven by rotation , Monatsh. Math., Volume 175 (2014), pp. 595-612 | Zbl | MR | DOI

Rosen, D. A Class of Continued Fractions Associated with Certain Properly Discontinuous Groups, Duke Math. J., Volume 21 (1954), pp. 549-563 | MR | Zbl | DOI

Schmidt, T. A. Remarks on the Rosen $λ$ -continued fractions, Number theory with an emphasis on the Markoff spectrum (Pollington, A.; Moran, W., eds.), Dekker, New York, 1993, pp. 227-238 | MR | Zbl

Series, C. The modular surface and continued fractions, J. London Math. Soc. (2), Volume 31 (1985), pp. 69-80 | MR | Zbl | DOI

Smillie, J.; Ulcigrai, C. Beyond Sturmian sequences: coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc. (3), Volume 2 (2011), pp. 291-340 | MR | Zbl | DOI

Vallée, B. Euclidean Dynamics, Discrete and Continuous Dynamical Systems – Series S, American Institute of Mathematical Sciences (2006), pp. 281-352 | MR | Zbl

Veech, W.A. Gauss measures for transformations on the space of interval exchange maps, Ann. Math., Volume 115 (1982), pp. 201-242 | MR | Zbl | DOI

Cité par Sources :