Related papers: Simultaneous dense and nondense orbits for noncomm…
We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for…
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is…
The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the…
We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full…
We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic…
Let $f: M \to M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with nondense forward orbit: $E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$ for some $y…
We say that a topological space $X$ is selectively highly divergent (SHD) if for every sequence of non-empty open sets $\{U_n\mid n\in\omega \}$ of $X$, we can find $x_n\in U_n$ such that the sequence $(x_n)$ has no convergent subsequences.…
For a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the space of quasimorphisms and quasi-cocycles on $N$ non-extendable to $G$. To treat this space, we establish the five-term exact sequence of cohomology relative to…
In this note we develop a half-space model for the pseudo-hyperbolic space $\mathbb{H}^{p,q}$, for any $p,q$ with $p\geq 1$. This half-space model embeds isometrically onto the complement of a degenerate totally geodesic hyperplane in…
We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of…
We show that each non-hyperbolic ergodic measure of a partially hyperbolic diffeomorphism on $\mathbb{T}^3$ which is homotopic to Anosov admits a full measure subset which intersects each center leaf in at most two points.
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove…
In this article, we extend Anderson's higher-dimensional Dehn filling construction to a large class of infinite-volume hyperbolic manifolds. This gives an infinite family of topologically distinct asymptotically hyperbolic Einstein…
We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a…
It's known from from work of Hofer, Wysocki, and Zehnder [1996] and Bourgeois [2002] that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at…
In this paper, we prove the existence of super-hyperbolic orbits in four-body problem, which solves a conjecture of Marchal-Saari. We also prove the existence of noncollision singularities in the same model, which solves a conjecture of…
In this paper we prove that for an ergodic hyperbolic measure $\omega$ of a $C^{1+\alpha}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $\omega$-full measured set $\widetilde{\Lambda}$ such that for every invariant…
We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost…
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $\Gamma$ on a topological manifold where the…
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…