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

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October 2013, 33(10): 4731-4742. doi: 10.3934/dcds.2013.33.4731

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, Poland

Received September 2010 Revised March 2013 Published April 2013

Abstract
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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.

Keywords: invariant measures, Expansion growth, entropy, transformatios groups..

Mathematics Subject Classification: Primary: 37B05; Secondary: 37B4.

Citation: Paweł G. Walczak. Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4731-4742. doi: 10.3934/dcds.2013.33.4731

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.

show all references

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.

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