arXiv subjects

Giovanni Forni

Publications and source records attributed to Giovanni Forni.

At least 37 records · Page 2Linked to original sources

arXiv2020

Ruelle resonances from cohomological equations

These notes are based on lectures given by the author at the Summer School on Teichmüller dynamics, mapping class groups and applications in Grenoble, France, in June 2018 and at the Oberwolfach Seminar on Anisotropic Spaces and their Applications to Hyperbolic and Parabolic Systems in June 2019. We derive results about the so-called Ruelle resonances and the asymptotics of correlations for several classes of systems from known results on cohomological equations and invariant distributions for the respective unstable vector fields. In particular, we consider pseudo-Anosov diffeomorphisms on surfaces of higher genus, for horocycle flows on surfaces of constant negative curvature and for partially hyperbolic automorphisms of Heisenberg 3-dimensional nilmanfolds. Ruelles resonances for pseudo-Anosov maps with applications to the cohomological equation for their unstable translation flows was recently studied in depth by F. Faure, S. Gouëzel and E. Lanneau [FGL] by methods based on the analysis of the transfer operator of the pseudo-Anosov map. Ruelle resonances for geodesic flows on hyperbolic compact manifolds of any dimension and of partially hyperbolic automorphisms of Heisenberg 3-dimensional nilmanfolds are studied by general results of Dyatlov, Faure and Guillarmou [DFG] and Faure and Tsujii [FT15] based on methods of semi-classical analysis. These works do not derive results on cohomological equations for unstable flows or horospherical foliations of these systems.

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arXiv2020

Limits of geodesic push-forwards of horocycle invariant measures

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous action of the Lie group SL(2, R) on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the SL(2,R)-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the "weak convergence" of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmueller geodesic flow.

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arXiv2020

Twisted cohomological equations for translation flows

We prove by methods of harmonic analysis a result on existence of solutions for twisted cohomological equations on translation surfaces with loss of derivatives at most 3+ in Sobolev spaces. As a consequence we prove that product translation flows on (3-dimensional) translation manifolds which are products of a (higher genus) translation surface with a (flat) circle are stable in the sense of A. Katok. In turn, our result on product flows implies a stability result of time-τ maps of translation flows on translation surfaces.

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arXiv2019

Lebesgue spectrum of countable multiplicity for conservative flows on the torus

We study the spectral measures of conservative mixing flows on the 2-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows.

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arXiv2018

Cohomological equation and local conjugacy class of Diophantine interval exchange maps

We extend some results of Marmi--Moussa--Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.

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arXiv2018

Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows

We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.

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arXiv2018

Mutliple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

We study time changes of bounded type Heisenberg nilflows $(ϕ_t)$ acting on the Heisenberg nilmanifold $M$. We show that for every positive $τ\in W^s(M)$, $s~>~7/2$, every non-trivial time change $(ϕ_t^τ)$ enjoys the Ratner property. As a consequence every mixing time change is mixing of all orders. Moreover we show that for every $τ\in W^s(M)$, $s>9/2$ and every $p,q\in \mathbb{N}$, $p\neq q$, $(ϕ_{pt}^τ)$ and $(ϕ_{qt}^τ)$ are disjoint. As a consequence Sarnak's Conjecture on Möbius disjointness holds for all such time changes.

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arXiv2017

Time-Changes of Heisenberg nilflows

We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions, have a modular property and scale exactly under the renor- malization dynamics. We then prove analyticity of the functionals in the transverse directions to the flow. As a consequence of this analyticity property we derive that there exists a full measure set of nilflows such that generic (non-trivial) time-changes are mixing and moreover have a "stretched polynomial" decay of correlations for sufficiently smooth functions. Moreover we also prove that there exists a full Hausdorff dimension set of nilflows such that generic non-trivial time-changes have polynomial decay of correlations.

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arXiv2017

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow.

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arXiv2015

Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups

We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index, this is the same as $SU(3,1)$ in its second exterior power representation. The example is constructed using origamis (i.e. square-tiled surfaces). It can be generalized to give monodromy inside the group $SO^*(2n)$ for all $n$, but in the general case the monodromy might split further inside the group. Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of Matheus-Yoccoz-Zmiaikou.

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arXiv2015

Invariant Distributions for homogeneous flows

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.

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arXiv2015

Effective equidistribution of twisted horocycle flows and horocycle maps

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms which are off the best known bounds of A. Good only by a logarithmic term. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.

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arXiv2014

On effective equidistribution for higher step nilflows

The main goal of this paper is to obtain optimal estimates on the speed of equidistribution of nilflows on higher step nilmanifolds. Under a Diophantine condition on the frequencies of the toral projection of the flow, we prove that for almost all points on the nilmanifold orbits become equidistributed at polynomial speed with exponent which decays quadratically as a function of the number of steps. The main novelty is the introduction of new techniques of renormalization (rescaling) in absence of a truly recurrent renormalization dynamics. Quantitative equidistribution estimates are derived from bounds on the scaling of invariant distributions (in Sobolev norms) and on the geometry of the nilmanifold under the rescaling.

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arXiv2014

Zero Lyapunov exponents of the Hodge bundle

By the results of G. Forni and of R. Treviño, the Lyapunov spectrum of the Hodge bundle over the Teichmüller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmüller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.

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arXiv2013

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich-Zorich cocycle, with respect to Masur-Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich-Zorich cocycle work as \emph{renormalization dynamics} for interval exchange transformations and translation flows. In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich-Zorich cocycle with respect to invariant measures other than the Masur-Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich-Zorich cocycle are very different from the case of Masur-Veech measures. Finally, we end these notes by constructing some examples of closed SL(2,R)-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary SL(2,R)-representations have arbitrarily small spectral gap (and in particular it has complementary series).

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arXiv2012

Lyapunov spectrum of invariant subbundles of the Hodge bundle

We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (or Kodaira--Spencer map) of the Hodge bundle with respect to Gauss--Manin connection and investigate the relations between the central {Oseldets} subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

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arXiv2012

Time-Changes of Horocycle Flows

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.

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arXiv2011

Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.

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