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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

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 & 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.  Google Scholar

[2]

M. Badura, Prescribing growth type of complete Riemannian manifolds of bounded geometry, Ann. Polon. Math., 75 (2000), 167-175.  Google Scholar

[3]

S. Banach, On Haar measure, Uspekhi Mat. Nauk, 2 (1936), 161-167. Google Scholar

[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.  Google Scholar

[5]

A. Candel and L. Conlon, "Foliations I," Amer. Math. Soc., Providence, 2000.  Google Scholar

[6]

S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Toulouse, 2 (1993), 15-52. doi: 10.5802/afst.756.  Google Scholar

[7]

R. Ellis, Distal transformation groups, Pacific J. Math., 24 (1957), 401-405. doi: 10.2140/pjm.1958.8.401.  Google Scholar

[8]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137.  Google Scholar

[9]

E. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 10.1007/BF02392274.  Google Scholar

[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.  Google Scholar

[11]

S. Matsumoto, The unique ergodicity of equicontinuous laminations, Hokkaido Math. J., 39 (2010), 389-403.  Google Scholar

[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.  Google Scholar

[13]

M. Rees, "On the Structure of Minimal Distal Transformation Groups with Topological Manifolds as Phase Spaces," thesis, University of Warwick, 1977. Google Scholar

[14]

W. Rudin, "Real and Complex Analysis," McGraw-Hill, New York, London etc., 1966.  Google Scholar

[15]

S. Saks, "Monografie Matematyczne," Theory of the Integral, 7, Warszawa - Lwów, 1937. Google Scholar

[16]

P. Walczak, "Dynamics of Foliations, Groups and Pseudogroups," Monografie Matematyczne, 64, Birkhäuser, Basel, 2004, doi: 10.1007/978-3-0348-7887-6.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. Badura, Prescribing growth type of complete Riemannian manifolds of bounded geometry, Ann. Polon. Math., 75 (2000), 167-175.  Google Scholar

[3]

S. Banach, On Haar measure, Uspekhi Mat. Nauk, 2 (1936), 161-167. Google Scholar

[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.  Google Scholar

[5]

A. Candel and L. Conlon, "Foliations I," Amer. Math. Soc., Providence, 2000.  Google Scholar

[6]

S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Toulouse, 2 (1993), 15-52. doi: 10.5802/afst.756.  Google Scholar

[7]

R. Ellis, Distal transformation groups, Pacific J. Math., 24 (1957), 401-405. doi: 10.2140/pjm.1958.8.401.  Google Scholar

[8]

H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515. doi: 10.2307/2373137.  Google Scholar

[9]

E. Ghys, R. Langevin and P. Walczak, Entropie géométrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 10.1007/BF02392274.  Google Scholar

[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.  Google Scholar

[11]

S. Matsumoto, The unique ergodicity of equicontinuous laminations, Hokkaido Math. J., 39 (2010), 389-403.  Google Scholar

[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.  Google Scholar

[13]

M. Rees, "On the Structure of Minimal Distal Transformation Groups with Topological Manifolds as Phase Spaces," thesis, University of Warwick, 1977. Google Scholar

[14]

W. Rudin, "Real and Complex Analysis," McGraw-Hill, New York, London etc., 1966.  Google Scholar

[15]

S. Saks, "Monografie Matematyczne," Theory of the Integral, 7, Warszawa - Lwów, 1937. Google Scholar

[16]

P. Walczak, "Dynamics of Foliations, Groups and Pseudogroups," Monografie Matematyczne, 64, Birkhäuser, Basel, 2004, doi: 10.1007/978-3-0348-7887-6.  Google Scholar

[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.  Google Scholar

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