Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T23:29:19.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of distal actions on locally compact groups

Published online by Cambridge University Press:  07 September 2017

C. R. E. RAJA  and
RIDDHI SHAH
C. R. E. RAJA
Affiliation:
Stat-Math Unit, Indian Statistical Institute (ISI), 8th Mile, Mysore Road, Bangalore 560 059, India email creraja@isibang.ac.in
RIDDHI SHAH
Affiliation:
School of Physical Sciences (SPS), Jawaharlal Nehru University (JNU), New Delhi 110 067, India email rshah@jnu.ac.in, riddhi.kausti@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We obtain a decomposition for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1340 - 1360
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abels, H.. Distal affine transformation groups. J. Reine Angew. Math. 299–300 (1978), 294–300.Google Scholar
Abels, H.. Distal automorphism groups of Lie groups. J. Reine Angew. Math. 329 (1981), 82–87.Google Scholar
Baumgartner, U. and Willis, G.. Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Israel J. Math. 142 (2004), 221–248.Google Scholar
Berend, D.. Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289(1) (1985), 393–407.Google Scholar
Berglund, J. F., Junghenn, H. D. and Milnes, P.. Analysis on semigroups. Function Spaces, Compactifications, Representations (Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication) . John Wiley, New York, 1989.Google Scholar
Conze, J. P. and Guivarc’h, Y.. Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris Ser. A 278 (1974), 1083–1086.Google Scholar
Dani, S. G. and Shah, R.. Contraction groups and semistable measure on p-adic Lie groups. Math. Proc. Cambridge Philos. Soc. 110 (1991), 299–306.Google Scholar
Dani, S. G. and Shah, R.. Contractible measures and Levy’s measures on Lie groups. Probability on Algebraic Structures (Contemporary Mathematics, 261) . Eds. Budzban, G., Feinsilver, P. and Mukherjea, A.. American Mathematical Society, Providence, RI, 2000, pp. 3–13.Google Scholar
Ellis, R.. Distal transformation groups. Pacific J. Math. 8 (1958), 401–405.Google Scholar
Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.Google Scholar
Guivarc’h, Y. and Raja, C. R. E.. Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces. Ergod. Th. & Dynam. Sys. 32 (2012), 1313–1349.Google Scholar
Hazod, W. and Siebert, E.. Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups. J. Theoret. Probab. 1 (1988), 211–225.Google Scholar
Hazod, W. and Siebert, E.. Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups. Structural Properties and Limit Theorems (Mathematics and its Applications, 531) . Kluwer Academic, Dordrecht, 2001.Google Scholar
Hewitt, E. and Ross, K. A.. Abstract harmonic analysis. Structure of Topological Groups, Integration Theory, Group Representations, Vol. I (Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 115) , 2nd edn. Springer, New York, 1979.Google Scholar
Iwasawa, K.. Some types of topological groups. Ann. of Math. (2) 50 (1949), 507–558.Google Scholar
Jaworski, W.. Contractive automorphisms of locally compact groups and the concentration function problem. J. Theoret. Probab. 10 (1997), 967–989.Google Scholar
Jaworski, W.. On contraction groups of automorphisms of totally disconnected locally compact groups. Israel J. Math. 172 (2009), 1–8.Google Scholar
Jaworski, W.. Contraction groups, ergodicity and distal properties of automorphisms on compact groups. Illinois J. Math. 56 (2012), 1023–1084.Google Scholar
Jaworksi, W. and Raja, C. R. E.. The Choquet–Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth. New York J. Math. 13 (2007), 159–174.Google Scholar
Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691–735.Google Scholar
Losert, V.. On the structure of groups with polynomial growth. Math. Z. 195 (1987), 109–117.Google Scholar
Losert, V.. On the structure of groups with polynomial growth. II. J. Lond. Math. Soc. (2) 63 (2001), 640–654.Google Scholar
Montgomery, D. and Zippin, L.. Topological Transformation Groups. Interscience, New York, 1955.Google Scholar
Raja, C. R. E.. On classes of p-adic Lie groups. New York J. Math. 5 (1999), 101–105.Google Scholar
Raja, C. R. E.. On growth, recurrence and the Choquet–Deny theorem for p-adic Lie groups. Math. Z. 251 (2005), 827–847.Google Scholar
Raja, C. R. E.. Distal actions and ergodic actions on compact groups. New York J. Math. 15 (2009), 301–318.Google Scholar
Raja, C. R. E.. On the existence of ergodic automorphisms in ergodic ℤ d -actions on compact groups. Ergod. Th. & Dynam. Sys. 30 (2010), 1803–1816.Google Scholar
Raja, C. R. E. and Shah, R.. Distal actions and shifted convolution property. Israel J. Math. 177 (2010), 391–412.Google Scholar
Rosenblatt, J.. A distal property of groups and the growth of connected locally compact groups. Mathematika 26(1) (1979), 94–98.Google Scholar
Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhauser, Basel, 1995.Google Scholar
Shah, R.. Orbits of distal actions on locally compact groups. J. Lie Theory 22 (2012), 587–599.Google Scholar
Siebert, E.. Contractive automorphisms on locally compact groups. Math. Z. 191 (1986), 73–90.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar