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Proceedings of the London Mathematical Society Advance Access originally published online on April 27, 2009
Proceedings of the London Mathematical Society 2009 99(3):658-696; doi:10.1112/plms/pdp013
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© 2009 London Mathematical Society

On the self-similarity problem for ergodic flows

K. Fraczek

Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
ul. Chopina 12/18
87-100 Torun
Poland
and
Institute of Mathematics
Polish Academy of Science
ul. Sniadeckich 8
00-956 Warszawa
Poland

M. Lemanczyk

Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
ul. Chopina 12/18
87-100 Torun
Poland
mlem@mat.uni.torun.pl

Received 27 July 2008. Revision received 22 January 2009.

Given an ergodic flow (Tt)t isin R we study the problem of its self-similarities, that is, we want to describe the set of s isin R for which the original flow is isomorphic to the flow (Tst)t isin R. The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self-similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self-similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.

2000 Mathematics Subject Classification 37A10 (primary), 47D03, 37E35 (secondary).

Research partially supported by MNiSzW grant N N201 384834 and Marie Curie ‘Transfer of Knowledge’ program, project MTKD-CT-2005-030042 (TODEQ).


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