We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate to the original action.
Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.
We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and same marked boundary distance, extending a result of Croke and Otal.
We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the X-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral rigidity of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish rigidity for almost all of them.
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study a family of generalized Bowen-Series boundary maps and prove the following rigidity result: in this family the topological entropy is constant and depends only on the genus of the surface. We give an explicit formula for this entropy and show that the value of the topological entropy also stays constant in the Teichmüller space of the surface. The proofs use conjugation to maps of constant slope.
Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichmüller space, taking all values ("flexibility") between zero…
Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},μ_g)$ with two base points. When $ G $ is exact, under a certain finiteness assumption on the measures $\{μ_g\}_{g\in G }$, we construct a boundary for the Bernoulli crossed product C$^*$-algebra that admits some commutativity and amenability in the sense of Ozawa's bi-exactness. As a consequence, we obtain that any such Bernoulli action is solid. This generalizes solidity of measure preserving Bernoulli actions by Ozawa and Chifan--Ioana, and is the first rigi…
In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $Γ$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $α$ be a smooth $Γ$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $ρ$ of $α$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $α$ and $ρ$, on a finite-index subgroup of $Γ$. If $α$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a coro…
It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the g…
