Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T05:43:00.911Z Has data issue: false hasContentIssue false
  • English
  • Français

On the self-similarity problem for smooth flows on orientable surfaces

Published online by Cambridge University Press:  16 September 2011

JOANNA KUŁAGA
JOANNA KUŁAGA*
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland (email: joanna.kulaga@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

On each compact connected orientable surface of genus greater than one we construct a class of flows without self-similarities.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1615 - 1660
Copyright
Copyright © Cambridge University Press 2011

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

[1]Ageev, O. N.. Nonsingular α-rigid maps. J. Dyn. Control Syst. 15(4) (2009), 449–452.CrossRefGoogle Scholar
[2]Arnol’d, V. I.. Topological and ergodic properties of closed 1-forms with incommensurable periods. Funktsional. Anal. i Prilozhen. 25(2) (1991), 1–12, 96.Google Scholar
[3]Boshernitzan, M.. A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52(3) (1985), 723–752.CrossRefGoogle Scholar
[4]Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G.. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245). Springer, New York, 1982.CrossRefGoogle Scholar
[5]Frączek, K. and Lemańczyk, M.. On the self-similarity problem for ergodic flows. Proc. Lond. Math. Soc. (3) 99(3) (2009), 658–696.CrossRefGoogle Scholar
[6]Frączek, K. and Lemańczyk, M.. On symmetric logarithm and some old examples in smooth ergodic theory. Fund. Math. 180(3) (2003), 241–255.CrossRefGoogle Scholar
[7]Frączek, K. and Lemańczyk, M.. On disjointness properties of some smooth flows. Fund. Math. 185(2) (2005), 117–142.CrossRefGoogle Scholar
[8]Frączek, K. and Lemańczyk, M.. On mild mixing of special flows over irrational rotations under piecewise smooth functions. Ergod. Th. & Dynam. Sys. 26(3) (2006), 719–738.CrossRefGoogle Scholar
[9]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[10]Helson, H. and Parry, W.. Cocycles and spectra. Ark. Mat. 16(2) (1978), 195–206.CrossRefGoogle Scholar
[11]Katok, A. and Thouvenot, J.-P.. Spectral properties and combinatorial constructions in ergodic theory. Handbook of Dynamical Systems, Vol. 1B. Elsevier B. V., Amsterdam, 2006, pp. 649–743.CrossRefGoogle Scholar
[12]Katok, A. B.. Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35(4) (1980), 301–310.CrossRefGoogle Scholar
[13]Keane, M.. Interval exchange transformations. Math. Z. 141 (1975), 25–31.CrossRefGoogle Scholar
[14]Kochergin, A. V.. Nonsingular saddle points and the absence of mixing. Mat. Zametki 19(3) (1976), 453–468 (in Russian).Google Scholar
[15]Marcus, B.. The horocycle flow is mixing of all degrees. Invent. Math. 46(3) (1978), 201–209.CrossRefGoogle Scholar
[16]Marmi, S., Moussa, P. and Yoccoz, J.-C.. The cohomological equation for Roth-type interval exchange maps. J. Amer. Math. Soc. 18(4) (2005), 823–872 (electronic).CrossRefGoogle Scholar
[17]Mayer, A.. Trajectories on the closed orientable surfaces. Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 71–84.Google Scholar
[18]Rauzy, G.. Échanges d’intervalles et transformations induites. Acta Arith. 34(4) (1979), 315–328.CrossRefGoogle Scholar
[19]Ryzhikov, V. V.. On a connection between the mixing properties of a flow with an isomorphism entering into its transformations. Mat. Zametki 49(6) (1991), 98–106, 159.Google Scholar
[20]Ryzhikov, V. V.. Stochastic intertwinings and multiple mixing of dynamical systems. J. Dynam. Control Systems 2(1) (1996), 1–19.CrossRefGoogle Scholar
[21]Ryzhikov, V. V.. Wreath products of tensor products, and a stochastic centralizer of dynamical systems. Mat. Sb. 188(2) (1997), 67–94.Google Scholar
[22]Danilenko, A. I. and Ryzhikov, V. V.. On self-similarities of ergodic flows. Preprint, 2010, arXiv:1011.0343.Google Scholar
[23]Sinai, Ya. G. and Ulcigrai, C.. Weak mixing in interval exchange transformations of periodic type. Lett. Math. Phys. 74(2) (2005), 111–133.CrossRefGoogle Scholar
[24]Ulcigrai, C.. Absence of mixing in area-preserving flows on surfaces. Ann. of Math. (2) 173(3) (2011), 1743–1778.CrossRefGoogle Scholar
[25]Veech, W. A.. Projective Swiss cheeses and uniquely ergodic interval exchange transformations. Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979–80) (Progress in Mathematics, 10). Birkhäuser, Boston, MA, 1981, pp. 113–193.Google Scholar
[26]Veech, W. A.. Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115(1) (1982), 201–242.CrossRefGoogle Scholar
[27]Veršik, A. M.. Multivalued mappings with invariant measure (polymorphisms) and Markov operators. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 72 (1977), 26–61, 223 (in Russian) (Engl. Transl. Many-valued measure-preserving mappings (polymorphisms) and Markovian operators. J. Math. Sci. 23(3) (1983), 2243–2266).Google Scholar
[28]Zorich, A.. Hamiltonian flows of multivalued hamiltonians on closed orientable surfaces. Unpublished, 1994. Available at http://www.mpim-bonn.mpg.de/preblob/4964.Google Scholar
[29]Zorich, A.. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46(2) (1996), 325–370.CrossRefGoogle Scholar
[30]Zorich, A.. Deviation for interval exchange transformations. Ergod. Th. & Dynam. Sys. 17(6) (1997), 1477–1499.CrossRefGoogle Scholar