Parabolic flows renormalized by partially hyperbolic maps

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.

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Fig. 1

Notes

  1. 1.

    In the reference the terminology “prequantum transfer operator for symplectic Anosov diffeomorphism” is used for this same object.

  2. 2.

    This equates to the choice of potential \(V= \ln \lambda \) in the reference [12].

  3. 3.

    This work fits into a diverse and ongoing line of research developing and using functional analytic methods for dynamical systems (see e.g., [1, 4,5,6,7,8, 10, 11, 13, 20, 21] and particularly [2]).

  4. 4.

    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.

  5. 5.

    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 \(N\).

  6. 6.

    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.

  7. 7.

    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.

  8. 8.

    We can compute explicitly the formula \(\exp {tW} = (\alpha t, \beta t, \frac{1}{2} \alpha \beta t^2)\).

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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.

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Butterley, O., Simonelli, L.D. Parabolic flows renormalized by partially hyperbolic maps. Boll Unione Mat Ital (2020). https://doi.org/10.1007/s40574-020-00235-8

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