Skip to main content Accessibility help
×
Home
Hostname: page-component-564cf476b6-4htn5 Total loading time: 0.222 Render date: 2021-06-20T17:23:56.563Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus

Published online by Cambridge University Press:  18 March 2021

VIVIANE BALADI
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation, CNRS, Sorbonne Université, Université de Paris, 4, Place Jussieu, 75005 Paris, France
Corresponding
E-mail address:

Abstract

We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$. We compare our results with very recent related work of Forni.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

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

References

Adam, A.. Horocycle averages on closed manifolds and transfer operators. Preprint, 2018, arXiv:1809.04062.Google Scholar
Baladi, V.. Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68). Springer, Cham, 2018.CrossRefGoogle Scholar
Baladi, V. and Tsujii, M.. Dynamical determinants and spectrum for hyperbolic diffeomorphisms. Geometric and Probabilistic Structures in Dynamics (Contemporary Mathematics, 469). American Mathematical Society, Providence, RI, 2008, pp. 2968.CrossRefGoogle Scholar
Carrand, J.. Logarithmic bounds for ergodic averages of constant type rotation number flows on the torus: a short proof. Preprint, 2020, arXiv:2012.07481.Google Scholar
Flaminio, L. and Forni, G.. Invariant distributions and time averages for horocycle flows. Duke Math. J. 119 (2003), 465526.Google Scholar
Forni, G.. On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces. Preprint, 2020, arXiv:2007.03144.Google Scholar
Giulietti, P. and Liverani, C.. Parabolic dynamics and anisotropic Banach spaces. JEMS 21 (2019), 27932858.CrossRefGoogle Scholar
Gouëzel, S. and Liverani, C.. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom. 79 (2008), 433477.CrossRefGoogle Scholar
Hasselblatt, B.. Regularity of the Anosov splitting and of horospheric foliations. Ergod. Th. & Dynam. Sys. 14 (1994), 645666.CrossRefGoogle Scholar
Herman, M.-R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5233.CrossRefGoogle Scholar
Hiraide, K.. A simple proof of the Franks–Newhouse theorem on codimension-one Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 21 (2001), 801806.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia Mathematics and Applications, 54). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
McCutcheon, R.. The Gottschalk–Hedlund theorem. Amer. Math. Monthly 106 (1999), 670672.CrossRefGoogle Scholar
Young, L.-S.. What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108 (2002), 733754.CrossRefGoogle Scholar

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.

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus
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.

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus
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.

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *