An analogue of the Variational Principle for group and pseudogroup actions
[Un analogue du principe variationnel pour les actions de groupe et pseudogroupe]
Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 839-863.

On généralise au cas des groupes d’homéomorphismes de type fini la notion d’entropie mesure locale introduite par Brin et Katok [7] pour une seule transformation. On applique la théorie des caractéristiques de type dimension d’un système dynamique élaborée par Pesin [25] pour obtenir une relation entre l’entropie topologique d’un pseudogroupe et d’un groupe d’homéomorphismes d’un espace métrique, définie par Ghys, Langevin et Walczak dans [12], et ses entropies mesure locale. On prouve un analogue du principe variationnel pour les actions de groupe et de pseudogroupe qui nous permet d’étudier les dynamiques locales des feuilletages.

We generalize to the case of finitely generated groups of homeomorphisms the notion of a local measure entropy introduced by Brin and Katok [7] for a single map. We apply the theory of dimensional type characteristics of a dynamical system elaborated by Pesin [25] to obtain a relationship between the topological entropy of a pseudogroup and a group of homeomorphisms of a metric space, defined by Ghys, Langevin and Walczak in [12], and its local measure entropies. We prove an analogue of the Variational Principle for group and pseudogroup actions which allows us to study local dynamics of foliations.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.5802/aif.2778
Classification : 37C85, 28D20, 37B40
Keywords: variational principle, topological entropy, Carathéodory structures, Carathéodory measures and dimensions, local measure entropy, pseudogroups, foliations, Hausdorff measure, homogeneous measure
Mot clés : principe variationnel, l’entropie topologique, structures Carathéodory, des mesures de Carathéodory et de dimensions de Carathéodory, l’entropie mesure locale, pseudogroups, feuilletages, mesure de Hausdorff, mesure homogene
Biś, Andrzej 1

1 University of Lodz Department of Mathematics and Computer Science ul. Banacha 22 90-238 Lodz (Poland)
@article{AIF_2013__63_3_839_0,
     author = {Bi\'s, Andrzej},
     title = {An analogue of the {Variational} {Principle} for group and pseudogroup actions},
     journal = {Annales de l'Institut Fourier},
     pages = {839--863},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {3},
     year = {2013},
     doi = {10.5802/aif.2778},
     mrnumber = {3137474},
     zbl = {1294.37011},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2778/}
}
TY  - JOUR
AU  - Biś, Andrzej
TI  - An analogue of the Variational Principle for group and pseudogroup actions
JO  - Annales de l'Institut Fourier
PY  - 2013
SP  - 839
EP  - 863
VL  - 63
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2778/
DO  - 10.5802/aif.2778
LA  - en
ID  - AIF_2013__63_3_839_0
ER  - 
%0 Journal Article
%A Biś, Andrzej
%T An analogue of the Variational Principle for group and pseudogroup actions
%J Annales de l'Institut Fourier
%D 2013
%P 839-863
%V 63
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2778/
%R 10.5802/aif.2778
%G en
%F AIF_2013__63_3_839_0
Biś, Andrzej. An analogue of the Variational Principle for group and pseudogroup actions. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 839-863. doi : 10.5802/aif.2778. https://aif.centre-mersenne.org/articles/10.5802/aif.2778/

[1] Álvarez López, J. A.; Candel, A. Equicontinuous foliated spaces, Math. Z., Volume 263 (2009) no. 4, pp. 725-774 | DOI | MR | Zbl

[2] Ambrosio, Luigi; Tilli, Paolo Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25, Oxford University Press, Oxford, 2004, viii+133 pages | MR | Zbl

[3] Biś, Andrzej Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst., Volume 11 (2004) no. 2-3, pp. 639-648 | DOI | MR | Zbl

[4] Biś, Andrzej; Urbański, Mariusz Some remarks on topological entropy of a semigroup of continuous maps, Cubo, Volume 8 (2006) no. 2, pp. 63-71 | MR | Zbl

[5] Bowen, Rufus Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., Volume 153 (1971), pp. 401-414 | DOI | MR | Zbl

[6] Bowen, Rufus Topological entropy for noncompact sets, Trans. Amer. Math. Soc., Volume 184 (1973), pp. 125-136 | DOI | MR | Zbl

[7] Brin, M.; Katok, A. On local entropy, Geometric dynamics (Rio de Janeiro, 1981) (Lecture Notes in Math.), Volume 1007, Springer, Berlin, 1983, pp. 30-38 | DOI | MR | Zbl

[8] Bufetov, A. Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Systems, Volume 5 (1999) no. 1, pp. 137-143 | DOI | MR | Zbl

[9] Coifman, Ronald R.; Weiss, Guido Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin, 1971, v+160 pages (Étude de certaines intégrales singulières) | MR | Zbl

[10] Falconer, Kenneth Techniques in fractal geometry, John Wiley & Sons Ltd., Chichester, 1997, xviii+256 pages | MR | Zbl

[11] Friedland, Shmuel Entropy of graphs, semigroups and groups, Ergodic theory of Z d actions (Warwick, 1993–1994) (London Math. Soc. Lecture Note Ser.), Volume 228, Cambridge Univ. Press, Cambridge, 1996, pp. 319-343 | DOI | MR | Zbl

[12] Ghys, É.; Langevin, R.; Walczak, P. Entropie géométrique des feuilletages, Acta Math., Volume 160 (1988) no. 1-2, pp. 105-142 | DOI | MR | Zbl

[13] Gromov, Misha Metric structures for Riemannian and non-Riemannian spaces, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007, xx+585 pages (Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates) | MR | Zbl

[14] Haefliger, André Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa (3), Volume 16 (1962), pp. 367-397 | Numdam | MR | Zbl

[15] Haefliger, André Groupoïdes d’holonomie et classifiants, Astérisque (1984) no. 116, pp. 70-97 Transversal structure of foliations (Toulouse, 1982) | MR | Zbl

[16] Haefliger, André Pseudogroups of local isometries, Differential geometry (Santiago de Compostela, 1984) (Res. Notes in Math.), Volume 131, Pitman, Boston, MA, 1985, pp. 174-197 | MR | Zbl

[17] Hajłasz, Piotr Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) (Contemp. Math.), Volume 338, Amer. Math. Soc., Providence, RI, 2003, pp. 173-218 | DOI | MR | Zbl

[18] Heinonen, Juha Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001, x+140 pages | DOI | MR | Zbl

[19] Luukkainen, Jouni; Saksman, Eero Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc., Volume 126 (1998) no. 2, pp. 531-534 | DOI | MR | Zbl

[20] Ma, Ji-Hua; Wen, Zhi-Ying A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 9-10, pp. 503-507 | DOI | MR | Zbl

[21] Mattila, Pertti Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995, xii+343 pages (Fractals and rectifiability) | MR | Zbl

[22] Mauldin, R. Daniel; Urbański, Mariusz Graph directed Markov systems, Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003, xii+281 pages (Geometry and dynamics of limit sets) | DOI | MR | Zbl

[23] Montgomery, Deane; Zippin, Leo Topological transformation groups, Interscience Publishers, New York-London, 1955, xi+282 pages | MR | Zbl

[24] Pesin, Yakov B. Dimension Type Characteristics for Invariant Sets of Dynamical Systems, Russian Math. Surveys, Volume 43 (1988), pp. 111-151 | DOI | MR | Zbl

[25] Pesin, Yakov B. Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997, xii+304 pages (Contemporary views and applications) | MR | Zbl

[26] Sumi, Hiroki Skew product maps related to finitely generated rational semigroups, Nonlinearity, Volume 13 (2000) no. 4, pp. 995-1019 | DOI | MR | Zbl

[27] Vol’berg, A .L.; Konyagin, S. V. There is a homogeneous measure on any compact subset of n , Soviet Math. Dokl., Volume 30 (1984), pp. 453-456 | Zbl

[28] Vol’berg, A .L.; Konyagin, S. V. On measure with the doubling condition, Math. USSR-Izv., Volume 30 (1988), pp. 629-638 | DOI | Zbl

[29] Walczak, P. Dynamics of Foliations, Groups and Pseudogroups, Monografie Matematyczne, 64, Birkhäuser, Basel, 2004 | MR | Zbl

Cité par Sources :