113 research outputs found
A note on Carnot geodesics in nilpotent Lie groups
We show that strictly abnormal geodesics arise in graded nilpotent Lie
groups. We construct such a group, for which some Carnot geodesics are strictly
abnormal; in fact, they are not normal in any subgroup. In the step-2 case we
also prove that these geodesics are always smooth. Our main technique is based
on the equations for the normal and abnormal curves, that we derive (for any
Lie group) explicitly in terms of the structure constants
Rhombic Tilings and Primordia Fronts of Phyllotaxis
We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of pri- mordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae
Ghost circles in lattice Aubry-Mather theory
Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice,
arise in Hamiltonian lattice mechanics as models for fe?rromagnetism and as
discretization of elliptic PDEs. Mathematically, they are a multidimensional
counterpart of monotone twist maps. They often admit a variational structure,
so that the solutions are the stationary points of a formal action function.
Classical Aubry-Mather theory establishes the existence of a large collection
of solutions of any rotation vector. For irrational rotation vectors this is
the well-known Aubry-Mather set. It consists of global minimizers and it may
have gaps.
In this paper, we study the gradient flow of the formal action function and
we prove that every Aubry-Mather set can be interpolated by a continuous
gradient-flow invariant family, the so-called "ghost circle". The existence of
ghost circles is first proved for rational rotation vectors and Morse action
functions. The main technical result is a compactness theorem for ghost
circles, based on a parabolic Harnack inequality for the gradient flow, which
implies the existence of ghost circles of arbitrary rotation vectors and for
arbitrary actions. As a consequence, we can give a simple proof of the fact
that when an Aubry-Mather set has a gap, then this gap must be parametrized by
minimizers, or contain a non-minimizing solution.Comment: 39 pages, 1 figur
Convergence in a Disk Stacking Model on the Cylinder
We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either finitely or infinitely many iterations. The infinite-time convergence is explained by a conserved quantity
Lagrangian Systems on Hyperbolic Manifolds
This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler-Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry-Mather theory of twist maps and the Hedlund-Morse-Gromov theory of minimal geodesies on closed surfaces and hyperbolic manifolds
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