Oscillating mushrooms: adiabatic theory for a non-ergodic system

Can elliptic islands contribute to sustained energy growth as parameters of a Hamiltonian system slowly vary with time? In this paper we show that a mushroom billiard with a periodically oscillating boundary accelerates the particle inside it exponentially fast. We provide an estimate for the rate of acceleration. Our numerical experiments confirms the theory. We suggest that a similar mechanism applies to general systems with mixed phase space.Comment: final revisio

An analytic Approach to Turaev's Shadow Invariant

In the present paper we extend the "torus gauge fixing approach" by Blau and Thompson (Nucl. Phys. B408(1):345--390, 1993) for Chern-Simons models with base manifolds M of the form M= \Sigma x S^1 in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M. We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined.Comment: 44 pages, 2 figures. Changes have been made in Sec. 2.3, Sec 2.4, Sec. 3.4, and Sec. 3.5. Appendix C is ne

Reshetikhin's Formula for the Jones Polynomial of a Link: Feynman diagrams and Milnor's Linking Numbers

We use Feynman diagrams to prove a formula for the Jones polynomial of a link derived recently by N.~Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large $k$ limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow us to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. We conjecture a relation between the dominant part of the phase and Milnor's linking numbers. We check it explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals.Comment: 33 pages, 11 figure

Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof of the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of three-dimensional Henon-like diffeomorphisms

Modified 6j-symbols and 3-manifold invariants

37 pages, 16 figuresInternational audienceWe show that the renormalized quantum invariants of links and graphs in the 3-sphere, derived from tensor categories in ["Modified quantum dimensions and re-normalized link invariants", arXiv:0711.4229] lead to modified 6j-symbols and to new state sum 3-manifold invariants. We give examples of categories such that the associated standard Turaev-Viro 3-manifold invariants vanish but the secondary invariants may be non-zero. The categories in these examples are pivotal categories which are neither ribbon nor semi-simple and have an infinite number of simple objects

Topological quantum field theory and invariants of graphs for quantum groups

On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the classical groups at simple even roots of unity provide examples of this construction. Calculational methods are developed which, in particular, yield the dimensions of the state spaces as well as a proof of the relation, previously announced for the case of $SU_q(2)$ by V.Turaev, between these models and corresponding ones based on the ribbon graph construction of Reshetikhin and Turaev.Comment: 38 page

Toward a general theory of linking invariants

Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2 disjoint. The classical linking number lk(phi_1,phi_2) is defined only when phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M). The affine linking invariant alk is a generalization of lk to the case where phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we constructed the first examples of affine linking invariants of nonzero-homologous spheres in the spherical tangent bundle of a manifold, and showed that alk is intimately related to the causality relation of wave fronts on manifolds. In this paper we develop the general theory. The invariant alk appears to be a universal Vassiliev-Goussarov invariant of order < 2. In the case where phi_1*[N_1] and phi_2*[N_2] are 0 in homology it is a splitting of the classical linking number into a collection of independent invariants. To construct alk we introduce a new pairing mu on the bordism groups of spaces of mappings of N_1 and N_2 into M, not necessarily under the restriction dim N_1 + dim N_2 +1 = dim M. For the zero-dimensional bordism groups, mu can be related to the Hatcher-Quinn invariant. In the case N_1=N_2=S^1, it is related to the Chas-Sullivan string homology super Lie bracket, and to the Goldman Lie bracket of free loops on surfaces.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper42.abs.htm

Simplified numerical form of universal finite type invariant of Gauss words

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.Comment: 12 pages, 3 table