Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms

Paweł G. Walczak

doi:10.3934/dcds.2013.33.4731

Article Contents

2013, Volume 33, Issue 10: 4731-4742. Doi: 10.3934/dcds.2013.33.4731

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Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms

Paweł G. Walczak^1,

1.
Katedra Geometrii, Wydział Matematyki i Informatyki, Uniwersytet Łódzki, Łódź

Received: September 2010

Revised: March 2013

Published: October 2013

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Abstract

We estimate expansion growth types (in the sense of Egashira) of certain distal groups of homeomorphisms and manifold diffeomorphisms.The estimate implies zero entropy (in the sense of Ghys, Langevin and the author) and existence of invariant measures for such groups. We prove also existence of invariant measures for pseudogroups satisfying some conditions of distality type.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 37B40.

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References

[1]	J. Alvarez Lopez and A. Candel, Equicontinuous foliated spaces, Math. Z., 263 (2009), 725-774.doi: 10.1007/s00209-008-0432-4.
[2]	M. Badura, Prescribing growth type of complete Riemannian manifolds of bounded geometry, Ann. Polon. Math., 75 (2000), 167-175.
[3]	S. Banach, On Haar measure, Uspekhi Mat. Nauk, 2 (1936), 161-167.
[4]	A. Biś and P. Walczak, Entropy of distal groups, pseudogroups and laminations, Ann. Polon. Math., 100 (2011), 45-54.doi: 10.4064/ap100-1-5.
[5]	A. Candel and L. Conlon, "Foliations I," Amer. Math. Soc., Providence, 2000.
[6]	S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Toulouse, 2 (1993), 15-52.doi: 10.5802/afst.756.
[7]	R. Ellis, Distal transformation groups, Pacific J. Math., 24 (1957), 401-405.doi: 10.2140/pjm.1958.8.401.
[8]	H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.doi: 10.2307/2373137.
[9]	E. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math., 160 (1988), 105-142.doi: 10.1007/BF02392274.
[10]	A. Haefliger, Foliations and compactly generated pseudogroups, in "Foliations: Geometry and Dynamics," Proc. of the Conf., Warsaw (2000), World Sci. Publ., Singapore (2002), 275-295.doi: 10.1142/9789812778246_0013.
[11]	S. Matsumoto, The unique ergodicity of equicontinuous laminations, Hokkaido Math. J., 39 (2010), 389-403.
[12]	W. Parry, Zero entropy of distal and related transformations, in "Topological Dynamics: An International Symposium" (eds. J. Auslander and W. H. Gottschalk), W. A. Benjamin, New York (1968), pp. 383-389.
[13]	M. Rees, "On the Structure of Minimal Distal Transformation Groups with Topological Manifolds as Phase Spaces," thesis, University of Warwick, 1977.
[14]	W. Rudin, "Real and Complex Analysis," McGraw-Hill, New York, London etc., 1966.
[15]	S. Saks, "Monografie Matematyczne," Theory of the Integral, 7, Warszawa - Lwów, 1937.
[16]	P. Walczak, "Dynamics of Foliations, Groups and Pseudogroups," Monografie Matematyczne, 64, Birkhäuser, Basel, 2004,doi: 10.1007/978-3-0348-7887-6.
[17]	A. Weil, "L'intégration dans les Groupes Topologiques et ses Applications," (French) [This book has been republished by the author at Princeton, N. J., 1941.], Actual. Sci. Ind., 869, Hermann et Cie, Paris, 1940.