Abstract
We consider nilflows on the Heisenberg nilmanifolds which are renormalized by partially hyperbolic automorphisms, i.e., parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeomorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.
Similar content being viewed by others
Notes
In the reference the terminology “prequantum transfer operator for symplectic Anosov diffeomorphism” is used for this same object.
This equates to the choice of potential \(V= \ln \lambda \) in the reference [12].
In [20, Rem. 2.15], it was written that the case of countable deviation spectrum corresponds to flows whereas the case of finite spectrum corresponds to maps. Instead we note here that the presence of a neutral direction (e.g., in the present work as well as when studying flows) is the distinguishing factor which determines the unavoidable presence of a countable number of eigenvalues in the problem.
The reference allows for higher dimension but here we restrict to the case (\(d=1\)) as required in this present work. Although the reference uses interchangeable \(N\) and \(\hbar \) where \(\hbar = \frac{1}{2\pi N} \) here we systematically use \(N\).
The primary reference [12] uses two operators, \(\mathcal {B}_{\hbar }\) and \(\mathcal {B}_{x}\). The first is defined as the Bargmann transform with a kernel similar to above (but slightly different scaling) and then the second is a scaling [12, (4.2.7)], of the first as \(\mathcal {B}_x := \tilde{\sigma }^{-1} \circ \mathcal {B}_{\hbar } \circ \sigma \) where \(\sigma h(x) := 2^{-\frac{1}{4}}h(2^{-\frac{1}{2}}x)\) and \(\tilde{\sigma }v(x,\xi ) := v(2^{-\frac{1}{2}}x, 2^{\frac{1}{2}}\xi )\). For our present purposes it makes sense to work directly with the required Bargmann transform which we denote \(\mathcal {B}_N\) and which corresponds precisely to \(\mathcal {B}_x\) of the reference.
Folland [16] refers to this as the symplectic Heisenberg group. It is also common to see the polarized Heisenberg group, given by the group law \((x,y,z) * (a,b,c) = (x+a, y+b, z+c+xb)\), which corresponds directly to matrix multiplication. The two groups are equivalent; the map \((x,y,z) \rightarrow (x,y,z + \frac{1}{2}xy) \) gives an isomorphism between them.
We can compute explicitly the formula \(\exp {tW} = (\alpha t, \beta t, \frac{1}{2} \alpha \beta t^2)\).
References
Adam, A.: Horocycle averages on closed manifolds and transfer operators. (2018) arXiv:1809.04062
Baladi, V.: The quest for the ultimate anisotropic Banach space (Corrections and complements. 170(2018), 1242–1247). J. Stat. Phys. 166, 525–557 (2017)
Baladi, V.: There are no deviations for the ergodic averages of the Giulietti–Liverani horocycle flows on the two-torus. Ergod. Theory Dyn. Syst. (2019) (To appear)
Bálint, P., Butterley, O., Melbourne, I.: Polynomial decay of correlations for flows, including Lorentz gas examples. Commun. Math. Phys. 368, 55–111 (2019)
Butterley, O.: An alternative approach to generalised \(BV\) and the application to expanding interval maps. Discrete Contin. Dyn. Syst. 33, 3355–3363 (2013)
Butterley, O.: Area expanding \(\cal{C}^{1+\alpha }\) suspension semiflows. Commun. Math. Phys. 325, 803–820 (2014)
Butterley, O.: A note on operator semigroups associated to chaotic flows. Ergod. Theory Dyn. Syst. 36, 1396–1408 (2016). (Corrigendum: 36 1409–1410)
Butterley, O., War, K.: Open sets of exponentially mixing Anosov flows. J. Eur. Math. Soc. 22, 2253–2285 (2020)
Faure, F.: Prequantum chaos: resonances of the prequantum cat map. J. Mod. Dyn. 1, 255–285 (2007)
Faure, F., Gouëzel, S., Lanneau, E.: Ruelle spectrum of linear pseudo-Anosov maps. J. l’École polytechnique - Mathématiques 6, 811–877 (2019)
Faure, F., Tsujii, M.: Band structure of the Ruelle spectrum of contact Anosov flows. C. P. Math. Acad. Sci. Paris 351(9–10), 385–391 (2013)
Faure, F., Tsujii, M.: Prequantum transfer operator for symplectic Anosov diffeomorphism. Astérisque v.375 Société Mathématique de France (2015)
Faure, F., Tsujii, M.: The semiclassical zeta function for geodesic flows on negatively curved manifolds. Invent. math. 208, 851–998 (2017)
Flaminio, L., Forni, G.: Invariant distributions and time averages for horocycle flows. Duke Math. J. 119, 465–526 (2003)
Flaminio, L., Forni, G.: Equidistribution of nilflows and applications to theta sums. Ergod. Theory Dyn. Syst. 26, 409–433 (2006)
Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Mathematics Studies. Princeton University Press, Princeton (1989)
Forni, G.: Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. Math. 146, 295–344 (1997)
Forni, G.: Asymptotic behavior of ergodic intergrals of renormalizable parabolic flows. In: Proceedings of the International Congress of Mathematicians August 20-28, 2002, vol. 3, pp. 317–326. Higher Education Press, Beijing (2002)
Forni, G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. 155, 1–103 (2002)
Giulietti, P., Liverani, C.: Parabolic dynamics and anisotropic Banach spaces. J. Eur. Math. Soc. 21, 2793–2858 (2019)
Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differ. Geom 79, 433–477 (2008)
Hasselblatt, B., Wilkinson, A.: Prevalence of non-Lipschitz Anosov foliations. Ergod. Theory Dyn. Syst. 19, 643–656 (1999)
Hurder, S., Katok, A.: Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Pub. Math. de l’IHÉS 72, 5–61 (1990)
Marmi, S., Moussa, P., Yoccoz, J.C.: The cohomological equation for Roth type interval exchange maps. J. Am. Math. Soc. 18, 823–872 (2005)
Palais, R.S., Stewart, T.E.: Torus bundles over a torus. Proc. Am. Math. Soc. 12, 26–29 (1961)
Acknowledgements
It is a pleasure to thank Giovanni Forni and Carlangelo Liverani for suggesting the study of this subject and for numerous useful discussions. O.B. thanks Centro di Ricerca Matematica Ennio De Giorgi for hospitality during the event “Renormalization in Dynamics” where the project was initiated. We are grateful to Alexander Adam, Viviane Baladi, Matías Delgadino, Sebastien Gouëzel, Minsung Kim, Davide Ravotti and Khadim War for several helpful discussions and comments. We thank Centre International de Rencontres Mathématiques for hospitality during the event “Probabilistic Limit Theorems for Dynamical Systems.” We thank Oberwolfach for hospitality during the event “Anisotropic Spaces and their Applications to Hyperbolic and Parabolic Systems” which facilitated key improvements of this text. O.B. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP 83C18000100006. O.B. was partially supported by the PRIN Grant “Regular and stochastic behaviour in dynamical systems” (PRIN 2017S35EHN). We are grateful to the anonymous referees for various comments which helped to improve this text.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Butterley, O., Simonelli, L.D. Parabolic flows renormalized by partially hyperbolic maps. Boll Unione Mat Ital 13, 341–360 (2020). https://doi.org/10.1007/s40574-020-00235-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40574-020-00235-8