Actions of semisimple Lie groups on circle bundles

Stationary measures for $\mathrm{SL}_2(\mathbb{R})$-actions on homogeneous bundles over flag varieties

Let $G$ be a real semisimple Lie group with finite centre and without compact factors, $Q<G$ a parabolic subgroup and $X$ a homogeneous space of $G$ admitting an equivariant projection on the flag variety $G/Q$ with fibres given by copies of lattice quotients of a semisimple factor of $Q$. Given a probability measure $μ$, Zariski-dense in a copy of $H=\mathrm{SL}_2(\mathbb{R})$ in $G$, we give a description of $μ$-stationary probability measures on $X$ and prove corresponding equidistribution results. Contrary to the results of Benoist-Quint corresponding to the case $G=Q$, the type of station…

Discriminant circle bundles over local models of Strebel graphs and Boutroux curves

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real periods; it appears naturally in the study of a neighbourhood of the Witten's cycle $W_1$ in the combinatorial model based on Jenkins-Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb R}_0([-3]^2)$ is t…

半单李群平稳作用的Furstenberg熵谱

Furstenberg entropy spectra of stationary actions of semisimple Lie groups

本文通过精细化Nevo-Zimmer射影因子定理并在平稳随机子群上构造Poisson丛，完整确定了SL(d,R)及其格的遍历平稳作用的Furstenberg熵谱。

Three-manifolds, Foliations and Circles, I

This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbolic 3-manifolds of every possible homological type. We show that all such foliations admit transverse pseudo-Anosov flows, and that in the universal cover of the hyperbolic cases, the leaves limit to sphere-filling Peano curves. The skew R-covered Anosov foliations of Sergio Fenl…

Breakdown of hyperbolicity for quasiperiodic attracting invariant circles in a family of three-dimensional Henon-like maps

We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasipe…

Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle

Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.

Hofer-Zehnder capacity of standard cotangent bundles

Let $M$ be a compact manifold with an effective semi-free circle action whose fixed point set has trivial normal bundle. We prove that its cotangent bundle endowed with the canonical symplectic form has bounded Hofer-Zehnder sensitive capacity. We give several examples like the product of any compact manifold with $§^n$ or a connected sum $\CP^n # ... # \CP^n$.