Skip to main content Accessibility help
×
Home
Hostname: page-component-5bf98f6d76-rtbc9 Total loading time: 0.35 Render date: 2021-04-20T23:56:32.920Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Fast and slow points of Birkhoff sums

Published online by Cambridge University Press:  11 July 2019

FRÉDÉRIC BAYART
Affiliation:
Université Clermont Auvergne, LMBP, UMR 6620 – CNRS, Campus des Cézeaux, 3 place Vasarely, TSA 60026, CS 60026 F-63178 Aubière Cedex, France Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117Budapest, Hungary email Frederic.Bayart@uca.fr, buczo@cs.elte.hu, Yanick.Heurteaux@uca.fr
ZOLTÁN BUCZOLICH
Affiliation:
Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117Budapest, Hungary email Frederic.Bayart@uca.fr, buczo@cs.elte.hu, Yanick.Heurteaux@uca.fr
YANICK HEURTEAUX
Affiliation:
Université Clermont Auvergne, LMBP, UMR 6620 – CNRS, Campus des Cézeaux, 3 place Vasarely, TSA 60026, CS 60026 F-63178 Aubière Cedex, France Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117Budapest, Hungary email Frederic.Bayart@uca.fr, buczo@cs.elte.hu, Yanick.Heurteaux@uca.fr

Abstract

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$ , where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$ . The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

Type
Original Article
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below.

References

Atkinson, G.. Recurrence of co-cycles and random walks. J. Lond. Math. Soc. (2) 13(3) (1976), 486488.CrossRefGoogle Scholar
Beck, J.. Randomness of the square root of 2 and the giant leap, Part 1. Period. Math. Hungar. 60(2) (2010), 137242.CrossRefGoogle Scholar
Beck, J.. Randomness of the square root of 2 and the giant leap, Part 2. Period. Math. Hungar. 62(2) (2011), 127246.CrossRefGoogle Scholar
Billingsley, P.. Probability and Measure (Wiley Series in Probability and Statistics). Wiley, New York, 1995.Google Scholar
Bromberg, M. and Ulcigrai, C.. A temporal central limit theorem for real-valued cocycles over rotations. Ann. Inst. Henri Poincaré Probab. Stat. 54(4) (2018), 23042334.CrossRefGoogle Scholar
Doob, J. L.. Stochastic Processes (Wiley Publications in Statistics). John Wiley & Sons Inc., New York; Chapman & Hall, Limited, London, 1953.Google Scholar
Eisner, T., Farkas, B., Haase, M. and Nagel, R.. Operator Theoretic Aspects of Ergodic Theory (Graduate Texts in Mathematics). Springer International Publishing, Cham, 2015.CrossRefGoogle Scholar
Fan, A. and Schmeling, J.. On fast Birkhoff averaging. Math. Proc. Cambridge Philos. Soc. 135 (2003), 443467.CrossRefGoogle Scholar
Fan, A. and Schmeling, J.. Everywhere divergence of one-sided ergodic Hilbert transform. Ann. Inst. Fourier (Grenoble) 68(6) (2018), 24772500.CrossRefGoogle Scholar
Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5233.CrossRefGoogle Scholar
Huveneers, F.. Subdiffusive behavior generated by irrational rotations. Ergod. Th. & Dynam. Sys. 29(4) (2009), 12171233.CrossRefGoogle Scholar
Kahane, J.-P.. Séries de Fourier Absolument Convergentes (Ergebnisse der Mathematik und Ihrer Grenzgebiete). Springer, Berlin, 1970.CrossRefGoogle Scholar
Kesten, H.. Uniform distribution mod 1. Ann. of Math. (2) 71 (1960), 445471.CrossRefGoogle Scholar
Kesten, H.. Uniform distribution mod 1. II. Acta Arith. 7 (1961/1962), 355380.CrossRefGoogle Scholar
Krengel, U.. On the speed of convergence in the ergodic theorem. Monatsh. Math. 86 (1978), 36.CrossRefGoogle Scholar
Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (Dover Books on Mathematics). Dover Publications, Mineola, NY, 2006.Google Scholar
Liardet, P. and Volný, D.. Continuous and differentiable functions in dynamical systems. Israel J. Math. 98 (1997), 2960.CrossRefGoogle Scholar
Rudin, W.. Fourier Analysis on Groups (Interscience Tracts in Pure and Applied Mathematics, 12). Interscience Publishers, New York, 1962.Google Scholar
Sinai, Y.. Topics in Ergodic Theory (Princeton Mathematical Series, 44). Princeton University Press, Princeton, NY, 1994.CrossRefGoogle Scholar
Zajiček, L.. Porosity and 𝜎-porosity. Real Anal. Exchange 13 (1987–1988), 314350.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 46 *
View data table for this chart

* Views captured on Cambridge Core between 11th July 2019 - 20th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Fast and slow points of Birkhoff sums
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Fast and slow points of Birkhoff sums
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Fast and slow points of Birkhoff sums
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *