arXiv subjects

Sven Sandfeldt

Publications and source records attributed to Sven Sandfeldt.

5 recordsLinked to original sources

arXiv2025

Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds

For a $\mathbb{R}^{k}-$action generated by vector fields $X_{1},...,X_{k}$ we define an operator $-(X_{1}^{2}+...+X_{k}^{2})$, the orbitwise laplacian. In this paper, we study and classify $\mathbb{R}^{k}-$actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is $1$. As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH $\mathbb{R}^{k}-$actions, proving, in particular, that it is always finite dimensional.

math.DS↗

arXiv2024

Global rigidity for some partially hyperbolic abelian actions with 1-dimensional center

We obtain a global rigidity result for abelian partially hyperbolic higher rank actions on certain $2-$step nilmanifolds $X_Γ$. We show that, under certain natural assumptions, all such actions are $C^{\infty}-$conjugated to an affine model. As a consequence, we obtain a centralizer rigidity result, classifying all possible centralizers for any $C^{1}-$small perturbation of an irreducible, affine partially hyperbolic map on $X_Γ$. Along the way, we also prove two results of independent interest. We describe fibered partially hyperbolic diffeomorphisms on $X_Γ$ and we show that topological conjugacies between partially hyperbolic actions and higher rank affine actions are $C^{\infty}$.

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arXiv2024

Centralizer classification and rigidity for some partially hyperbolic toral automorphisms

In this paper we consider local centralizer classification and rigidity of some toral automorphisms. In low dimensions we classify up to finite index possible centralizers for volume preserving diffeomorphisms $f$ $C^{1}-$close to an ergodic irreducible toral automorphism $L$. Moreover, we show a rigidity result in the case that the centralizer of $f$ is large: If the smooth centralizer $Z^{\infty}(f)$ is virtually isomorphic to that of $L$ then $f$ is $C^{\infty}-$conjugate to $L$. In higher dimensions we show a similar rigidity result for certain irreducible toral automorphisms. We also classify up to finite index all possible centralizers for symplectic diffeomorphisms $C^{5}-$close to a class of irreducible symplectic automorphisms on tori of any dimension.

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arXiv2024

Partially hyperbolic lattice actions on 2-step nilmanifolds

We prove global rigidity results for actions of higher rank lattices on nilmanifolds containing a partially hyperbolic element. We consider actions of higher rank lattices on tori or on $2-$step nilpotent nilmanifolds, such that the actions contain a partially hyperbolic element with $1-$dimensional center. In this setting we prove, under a technical assumption on the partially hyperbolic element, that any such action must be by affine maps. This extends results by Brown, Rodriguez Hertz, and Wang to certain lattice actions that are not Anosov.

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arXiv2021

Bounds of the spectral radius of the induced map on cohomology

In this paper we study the relationship between Lyapunov exponents and the induced map on cohomology for $C^{1}-$diffeomorphisms on compact manifolds. We show that if the induced map on cohomology has spectral radius strictly larger than 1, then the diffeomorphism has an invariant ergodic measure with at least one positive Lyapunov exponent. Furthermore, if the diffeomorphism also preserves a continuous volume form then it has an invariant ergodic measure with at least one positive and one negative Lyapunov exponent, in agreement with Shub's entropy conjecture. We also consider diffeomorphisms preserving a measure equivalent to volume. In this case we show that if the Lyapunov metric satisfies an integrability condition and the induced map on cohomology has spectral radius strictly larger than 1, then the diffeomorphism has non-zero Lyapunov exponents on a set of positive volume.

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