Related papers: Farey neighbors and hyperbolic Lorenz knots

Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables…

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume…

Let $M$ be a closed, negatively curved Riemannian manifold of dimension $n \neq 4, 8$ with strictly $1/4$-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with…

We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…

We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2,R), the determination…

The shortening of bodies in the direction of motion, Lorentz contraction, follows from the solution of Maxwell's equations. Moving light clocks will tick slower than those at rest because the speed of light does not depend on a source of…

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is…

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has…

We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve the product ordering of R^n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the…

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum…

A knotted surface in the 4-sphere may be described by means of a hyperbolic diagram that captures the 0-section of a special Morse function, called a hyperbolic decomposition. We show that every hyperbolic decomposition of a knotted surface…

We comment on mathematical results about the statistical behavior of Lorenz equations an its attractor, and more generally to the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be…

We classify the unitary, renormalizable, Lorentz violating quantum field theories of interacting scalars and fermions, obtained improving the behavior of Feynman diagrams by means of higher space derivatives. Higher time derivatives are not…

We investigate the quantitative and analytic aspects of the near-parabolic renormalization scheme introduced by Inou and Shishikura in 2006. These provide techniques to study the dynamics of some holomorphic maps of the form $f(z) = e^{2\pi…

The discovery of polynomial invariants of knots and links, ignited by V. F. R. Jones, leads to the formulation of polynomial invariants of spatial graphs. The Yamada polynomial, one of such invariants, is frequently utilized for practical…

The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial…

We show that any neighborhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on $N$-symbols for all $N\geq 2$. This will be achieved by showing switching associated with networks of…

We study the dynamics of the renormalization operator acting on the space of pairs (v,t), where v is a diffeomorphism and t belongs to [0,1], interpreted as unimodal maps x-->v(q_t(x)), where q_t(x)=-2t|x|^a+2t-1. We prove the so called…

We prove various new trigonometric and hyperbolic inequalities of Jordan, Wilker, Huygens or Cusa-Huygens type. Connections with bivariate means, as well as monotonicity and convexity properties are pointed out, too.

We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.