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Some properties of distal actions on locally compact groups

  • C. R. E. RAJA (a1) and RIDDHI SHAH (a2)


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.



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[1] Abels, H.. Distal affine transformation groups. J. Reine Angew. Math. 299–300 (1978), 294300.
[2] Abels, H.. Distal automorphism groups of Lie groups. J. Reine Angew. Math. 329 (1981), 8287.
[3] Baumgartner, U. and Willis, G.. Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Israel J. Math. 142 (2004), 221248.
[4] Berend, D.. Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc. 289(1) (1985), 393407.
[5] 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.
[6] Conze, J. P. and Guivarc’h, Y.. Remarques sur la distalité dans les espaces vectoriels. C. R. Acad. Sci. Paris Ser. A 278 (1974), 10831086.
[7] Dani, S. G. and Shah, R.. Contraction groups and semistable measure on p-adic Lie groups. Math. Proc. Cambridge Philos. Soc. 110 (1991), 299306.
[8] 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. 313.
[9] Ellis, R.. Distal transformation groups. Pacific J. Math. 8 (1958), 401405.
[10] Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477515.
[11] 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), 13131349.
[12] 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), 211225.
[13] 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.
[14] 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.
[15] Iwasawa, K.. Some types of topological groups. Ann. of Math. (2) 50 (1949), 507558.
[16] Jaworski, W.. Contractive automorphisms of locally compact groups and the concentration function problem. J. Theoret. Probab. 10 (1997), 967989.
[17] Jaworski, W.. On contraction groups of automorphisms of totally disconnected locally compact groups. Israel J. Math. 172 (2009), 18.
[18] Jaworski, W.. Contraction groups, ergodicity and distal properties of automorphisms on compact groups. Illinois J. Math. 56 (2012), 10231084.
[19] 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), 159174.
[20] Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691735.
[21] Losert, V.. On the structure of groups with polynomial growth. Math. Z. 195 (1987), 109117.
[22] Losert, V.. On the structure of groups with polynomial growth. II. J. Lond. Math. Soc. (2) 63 (2001), 640654.
[23] Montgomery, D. and Zippin, L.. Topological Transformation Groups. Interscience, New York, 1955.
[24] Raja, C. R. E.. On classes of p-adic Lie groups. New York J. Math. 5 (1999), 101105.
[25] Raja, C. R. E.. On growth, recurrence and the Choquet–Deny theorem for p-adic Lie groups. Math. Z. 251 (2005), 827847.
[26] Raja, C. R. E.. Distal actions and ergodic actions on compact groups. New York J. Math. 15 (2009), 301318.
[27] Raja, C. R. E.. On the existence of ergodic automorphisms in ergodic ℤ d -actions on compact groups. Ergod. Th. & Dynam. Sys. 30 (2010), 18031816.
[28] Raja, C. R. E. and Shah, R.. Distal actions and shifted convolution property. Israel J. Math. 177 (2010), 391412.
[29] Rosenblatt, J.. A distal property of groups and the growth of connected locally compact groups. Mathematika 26(1) (1979), 9498.
[30] Schmidt, K.. Dynamical Systems of Algebraic Origin (Progress in Mathematics, 128) . Birkhauser, Basel, 1995.
[31] Shah, R.. Orbits of distal actions on locally compact groups. J. Lie Theory 22 (2012), 587599.
[32] Siebert, E.. Contractive automorphisms on locally compact groups. Math. Z. 191 (1986), 7390.
[33] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.


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