Related papers: Measure rigidity beyond uniform hyperbolicity: Inv…

We prove that the restriction of a probability measure invariant under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group to the leaves of the central foliation is severely restricted. We also…

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed…

We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with…

We give a complete solution to the local classification program of higher rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity of abelian ergodic algebraic actions for symmetric space examples, twisted symmetric…

We prove that any real-analytic action of $SL(n,\Z), n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard…

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some…

We show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This…

The purpose of this paper is to describe some conjectures and results on the existence and uniqueness of invariant measures on formal completions of Kac-Moody groups and associated homogeneous spaces. Existence is rigorously established in…

In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{H\"older}}$…

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher…

For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for…

We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems.…

Let $\mathscr{F}=(M,\mathscr{L},E)$ be a Brody-hyperbolic singular holomorphic foliation on a compact complex manifold $M$. Suppose that $\mathscr{F}$ has isolated singularities and that its Poincar\'e metric is complete. This is the case…

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, an orientation-preserving leafwise…

We define "slow" entropy invariants for Z^2 actions on infinite measure spaces, which measures growth of itineraries at subexponential scales. We use this to construct infinite-measure preserving Z^2 actions which cannot be realized as a…

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image…

If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving $CC^{1+a}$, $a>0$,…

We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with…

We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for…

We show that the harmonic measure on a product of boundaries satisfies dimension conservation for a random walk with non-elementary marginals on a countable group acting on a product of hyperbolic spaces under the finite first moment…