Searcharxiv⌕ Search

arXiv subjects

Danijela Damjanovic

Publications and source records attributed to Danijela Damjanovic.

13 recordsLinked to original sources
arXiv2025

The Zimmer Program for partially hyperbolic actions

Zimmer's superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the {\it Zimmer program}. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin. We make progress in the program by showing smooth conjugacy to a bi-homogeneous model (up to a finite cover) for volume-preserving actions of semisimple Lie groups without compact or rank one factors, which have two key assumptions: partial hyperbolicity for a large class of elements ({\it totally partial hyperbolicity}) and accessibility, a condition on the webs generated by dynamically-defined foliations. We also obtain classification for actions of higher-rank abelian groups satisfying stronger assumptions.

math.DS↗
arXiv2023

Transitive centralizers and fibered partially hyperbolic systems

We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension $f_0\colon M\to M$ of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any $f\in \mathrm{Diff}^\infty(M)$ sufficiently $C^1$-close to $f_0$ has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then $f$ is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we further classify the centralizer of any partially hyperbolic diffeomorphism on a $3$-dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually $\mathbb Z\times \mathbb T$.

math.DS↗
arXiv2023

KAM-rigidity for parabolic affine abelian actions

We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $ρ_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $ρ_L$ has a $\mathbb Z$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine $\mathbb Z^2$-action with linear part $ρ_L$ is KAM-rigid under volume preserving perturbations.

math.DS↗
arXiv2022

$C^1$ actions on manifolds by lattices in Lie groups

In this paper we study Zimmer's conjecture for $C^1$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in $SL(n, \R)$, the dimensional bound is sharp.

math.DS↗
arXiv2020

Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with $1$-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -- including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually $\mathbb Z^\ell$ or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered in [2,3] for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25]. We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.

math.DS↗
arXiv2020

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $\mathbb R^2$ actions on homogeneous spaces $G/Γ$, with at least one quasi-unipotent generator, where $G= SL(n, \mathbb R)$. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

math.DS↗
arXiv2018

On classification of higher rank Anosov actions on compact manifold

We prove global smooth classification results for TNS totally Anosov Z^k actions on general compact manifolds, under each one of the following conditions: joint integrability, resonance-free or Lyapunov pinching condition. Unlike the previous results, we do not require any uniform quasiconformality or pinching condition of action elements on coarse Lyapunov distributions, nor do we have any restriction on the dimension of coarse Lyapunov distributions. The main novelty is in proving a new standard form of the derivative cocycle for any TNS totally Anosov Z^k action on general manifold. A main idea is to create a new mechanism called a non-uniform redefining argument to prove continuity of general dynamically-defined object, which should apply to more general rigidity problems in dynamical systems.

math.DS↗
arXiv2017

On conservative partially hyperbolic abelian actions with compact center foliation

We consider smooth partially hyperbolic volume preserving Z^k actions on smooth manifolds, with uniformly compact center foliation. We show that under certain irreducibility condition on the action, bunching and uniform quasiconformality conditions, the action is a smooth fiber bundle extension of an Anosov action, or the center foliation is pathological. We obtain several corollaries of this result. For example, we prove a global dichotomy result that any smooth conservative circle extension over a maximal Cartan action is either essentially a product of an action by rotations and a linear Anosov action on the torus, or has a pathological center foliation.

math.DS↗
arXiv2017

Diffeomorphism group valued cocycles over higher rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over higher rank abelian actions, with values in diffeomorphism groups, which does not require any restrictions on the smallness of the cocycle, nor on the diffeomorphism group.

math.DS↗
arXiv2014

Cocycle Rigidity and Splitting for some Discrete Parabolic Actions

We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb Z^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/Γ$, where the lattice $Γ$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.

math.DS↗
arXiv2013

KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.

math.DS↗
arXiv2013

Actions with globally hypoelliptic leafwise Laplacian and rigidity

We prove several results concerning smooth $\mathbb R^k$ actions with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and cohomology is often finite-dimensional, even trivial. Further we consider a class of examples of $\mathbb R^2$ actions on 2-step nilmanifolds, which have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.

math.DS↗