On the cohomological equation for nilflows
Livio Flaminio Giovanni Forni
Let $X$ be a vector field on a compact connected manifold $M$. An important question in dynamical systems is to know when a function $g: M\to \mathbb{R}$ is a coboundary for the flow generated by $X$, i.e., when there exists a function $f: M\to \mathbb{R}$ such that $Xf=g$. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions $D_n$ such that any sufficiently smooth function $g$ is a coboundary iff it belongs to the kernel of all the distributions $D_n$.
keywords: Nilflows Cohomological Equations.
Linearization of cohomology-free vector fields
Livio Flaminio Miguel Paternain
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
keywords: Cohomological Equations Greenfield-Wallach and Katok conjectures.
Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds
Salvatore Cosentino Livio Flaminio
We prove quantitative equidistribution results for actions of Abelian subgroups of the $(2g+1)$-dimensional Heisenberg group acting on compact $(2g+1)$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the classical results of Hardy and Littlewood [25,26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.
keywords: cohomological equation. Equidistribution Heisenberg group
Invariant distributions for homogeneous flows and affine transformations
Livio Flaminio Giovanni Forni Federico Rodriguez Hertz
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.
keywords: Cohomological equations homogeneous flows.

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