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Maximizing points and coboundaries for an irrational rotation on a circle

Published online by Cambridge University Press:  16 January 2012

JULIEN BRÉMONT  and
ZOLTÁN BUCZOLICH
JULIEN BRÉMONT
Affiliation:
LAMA, Université Paris-Est, 61, avenue du Général de Gaulle, 94010 Créteil cedex, France (email: [email protected])
ZOLTÁN BUCZOLICH
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117 Budapest, Hungary (email: [email protected])
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Abstract

Consider an irrational rotation of a unit circle and a real continuous function. A point is declared ‘maximizing’ if the growth of the ergodic sums at this point is maximal up to an additive constant. In the case of two-sided ergodic sums, the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build, for the ‘usual’ one-sided ergodic sums, examples in Hölder or smooth classes, indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but also for the transformation 2x mod 1. For this latter transformation, it is known that any Hölder continuous function has a maximizing point.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 24 - 48
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Aaronson, J., Lemańczyk, M., Mauduit, C. and Nakada, H.. Koksma’s inequality and group extensions of Kronecker transformations. Algorithmics, Dynamics and Fractals. Ed. Takahashi, Y.. Plenum Press, New York, 1995.Google Scholar
[2]Arnold, V.. Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21–86.Google Scholar
[3]Bousch, T.. Le Poisson n’a pas d’arêtes. Ann. Inst. Henri Poincaré Probab. Stat. 36 (2000), 489–508.CrossRefGoogle Scholar
[4]Bousch, T.. Le lemme de Mané–Conze–Guivarc’h pour les systèmes amphidynamiques rectifiables. Ann. Fac. Sci. Toulouse Math. (6) 20(1) (2011), 1–14.CrossRefGoogle Scholar
[5]Bousch, T. and Jenkinson, O.. Cohomology classes of dynamically non-negative C k functions. Invent. Math. 148(1) (2002), 207–217.CrossRefGoogle Scholar
[6]Brémont, J.. Ergodic non-abelian smooth extensions of an irrational rotation. J. Lond. Math. Soc. 81 (2010), 457–476.CrossRefGoogle Scholar
[7]Browder, F. E.. On the iteration of transformations in noncompact minimal dynamical systems. Proc. Amer. Math. Soc. 9 (1958), 773–780.CrossRefGoogle Scholar
[8]Carlson, D. A., Haurie, A. B. and Leizarowitz, A.. Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer, 1991.CrossRefGoogle Scholar
[9]Conze, J.-P.. Recurrence, ergodicity and invariant measures for cocycles over a rotation. Contemp. Math. 485 (2009), 45–70.CrossRefGoogle Scholar
[10]Conze, J.-P. and Guivarc’h, Y.. Croissance des sommes ergodiques et principe variationnel, unpublished manuscript, Rennes I.Google Scholar
[11]Cohen, A. and Conze, J.-P.. Régularité des bases d’ondelettes et mesures ergodiques. Rev. Mat. Iberoam. 8(3) (1992).CrossRefGoogle Scholar
[12]Feldman, J. and Moore, C.. Ergodic equivalence relations, cohomology and von Neumann algebras, I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.CrossRefGoogle Scholar
[13]Gottschalk, W. and Hedlund, G.. Topological Dynamics. Amer. Math. Soc. Colloq. Publ. 36 (1955).Google Scholar
[14]Jenkinson, O.. Ergodic Optimization. Discrete Contin. Dyn. Syst. 15 (2006), 197–224.CrossRefGoogle Scholar
[15]Khinchin, A. Ya.. Continued Fractions. The University of Chicago Press, Chicago, IL, 1935.Google Scholar
[16]Kraaikamp, C. and Liardet, P.. Good approximations and continued fractions. Proc. Amer. Math. Soc. 112(2) (1991), 303–309.CrossRefGoogle Scholar
[17]Lagarias, J. C. and Wang, Y.. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214 (1995), 17–42.CrossRefGoogle Scholar
[18]Lemańczyk, M., Parreau, F. and Volný, D.. Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces. Trans. Amer. Math. Soc. 348(12) (1996), 4919–4938.CrossRefGoogle Scholar
[19]Lin, M. and Sine, R.. Ergodic theory and the functional equation (I−T)x=y. J. Operator Theory 10(1) (1983), 153–166.Google Scholar
[20]Mané, R.. Generic properties of problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996), 273–310.CrossRefGoogle Scholar
[21]Rényi, A.. Probability Theory (North-Holland Series in Applied Mathematics and Mechanics, 10). North-Holland, Amsterdam–London, 1970; German version 1962, French version 1966, new Hungarian edition 1965.Google Scholar
[22]Schmidt, K.. Lectures on Cocycles of Ergodic Transformations Groups (Lecture Notes in Mathematics, 1). MacMillan Co. of India, 1977.Google Scholar