Related papers: Chaos and integrability in SL(2,R)-geometry
In the mid-1980's, M. Gromov used his machinery of the $h$-principle to prove that there exists totally real embeddings of $S^3$ into $\mathbb{C}^3$. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real…
In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive…
We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general…
Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $\Theta$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We…
We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore…
Given a closed hyperbolic 3-manifold M with a quasigeodesic flow we construct a \pi_1-equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal P to the lifted flow on H^3 has…
We identify the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $4$ and and element of order $p\geq 2$. Thus $\Gamma$ has a…
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry,…
In this paper we study the behavior of geodesics on cones over arbitrary $C^3$-smooth closed Riemannian manifolds. We show that the geodesic flow on such cones admits first integrals whose values uniquely determine almost all geodesics…
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schr\"odinger equation, the complex…
We consider a $\pi_{1}$--injective immersion $f:\Sigma\to M$ from a compact surface $\Sigma$ to a hyperbolic 3--manifold $M$. Let $\Gamma$ denote the copy of $\pi_{1}\Sigma$ in $\mathrm{Isom}({\mathbb{H}}^{3})$ induced by the immersion and…
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group…
The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms…
We include a Chern-Simons term in a GL(3,R) gauge formulation of gravity with a cosmological contribution in 2+1 dimension and we explore consistence showing that excitations must be causal and standard topological massive gravity is…
We investigate the presence of topological structures and multiple phase transitions in the O(3)-sigma model with the gauge field governed by Maxwell's term and subject to a so-called Gausson's self-dual potential. To carry out this study,…
We propose a new condition $\aleph$ which enables to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov's theorem on non-integrability on surfaces of higher genus.…
We study surface groups $\Gamma$ in $SO(4,1)$, which is the group of Mobius tranformations of $S^3$, and also the group of isometries of $\mathbb{H}^4$. We consider such $\Gamma$ so that its limit set $\Lambda_\Gamma$ is a quasi-circle in…
We construct classes of two-dimensional aperiodic Lorentz systems that have infinite horizon and are 'chaotic', in the sense that they are (Poincar\'e) recurrent, uniformly hyperbolic, ergodic, and the first-return map to any scatterer is…
Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts…
A Fuchsian group $\Gamma$ has a modular embedding if its adjoint trace field is a totally real number field and every unbounded Galois conjugate $\Gamma^\sigma$ comes equipped with a holomorphic (or conjugate holomorphic) map ${\phi^\sigma…