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On the rank of von Neumann special flows

Published online by Cambridge University Press:  14 March 2017

ADAM KANIGOWSKI  and
ANTON V. SOLOMKO
ADAM KANIGOWSKI
Affiliation:
Department of Mathematics, Pennsylvania State University, McAllister Building, 54 McAllister St, PA 16801, USA email [email protected]
ANTON V. SOLOMKO
Affiliation:
University of Bristol, School of Mathematics, Howard House, Queens Avenue, Bristol BS8 1SN, UK email [email protected]
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Abstract

We prove that special flows over an ergodic rotation of the circle under a $C^{1}$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2245 - 2256
Copyright
© Cambridge University Press, 2017 

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References

Chacon, R. V.. Approximation and spectral multiplicity. Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970). Springer, Berlin, 1970, pp. 18–27.Google Scholar
Fayad, B.. Rank one and mixing differentiable flows. Invent. Math. 160 (2005), 305–340.Google Scholar
Ferenczi, S.. Systems of finite rank. Colloq. Math. 73 (1997), 35–65.Google Scholar
Fayad, B. and Kanigowski, A.. On multiple mixing for a class of conservative surface flows. Invent. Math. 203 (2016), 555–614.Google Scholar
Fra̧czek, K. and Lemańczyk, M.. A class of special flows over irrational rotations which is disjoint from mixing flows. Ergod. Th. & Dynam. Sys. 24 (2004), 1083–1095.Google Scholar
Fra̧czek, K. and Lemańczyk, M.. On mild mixing of special flows over irrational rotations under piecewise smooth functions. Ergod. Th. & Dynam. Sys. 26 (2006), 719–738.Google Scholar
Iwanik, A., Lemańczyk, M. and Mauduit, C.. Piecewise absolutely continuous cocycles over irrational rotations. J. Lond. Math. Soc. (2) 59 (1999), 171–187.Google Scholar
Kanigowski, A.. Slow entropy for smooth flows on surfaces. Preprint.Google Scholar
Katok, A.. Cocycles, cohomology and combinatorial constructions in ergodic theory. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) . American Mathematical Society, Providence, RI, 2001, pp. 107–173. In collaboration with E. A. Robinson, Jr.Google Scholar
Khinchin, A. Ya.. Continued Fractions. University of Chicago Press, Chicago, 1964.Google Scholar
Kanigowski, A., Kułaga-Przymus, J. and Ulcigrai, C.. Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces. Preprint, 2016 arXiv:1606.09189.Google Scholar
Kočergin, A. V.. On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus. Dokl. Akad. Nauk SSSR 205 (1972), 949–952.Google Scholar
Kanigowski, A. and Solomko, A. V.. On isomorphism problem for von Neumann flows with one discontinuity. Preprint, 2016, arXiv:1610.02712.Google Scholar
Ratner, M.. Horocycle flows, joinings and rigidity of products. Ann. of Math. (2) 118 (1983), 277–313.Google Scholar
von Neumann, J.. Zur Operatorenmethode in der klassischen Mechanik. Ann. of Math. (2) 33 (1932), 587–642.Google Scholar