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Search my papers in the AMS database. All papers listed below are authored by L. Butler unless noted otherwise. Abstracts of papers and preprints. If you have problems with the links below, click on the link following the DOI number or go to DOI.org and enter the DOI listed along with the citation.
Abstracts:
ABSTRACT.This paper studies properties of Tonelli Hamiltonian systems that possess n independent but not necessarily involutive constants of motion. We obtain results reminiscent of the Liouville-Arnold theorem under suitable hypothesis on the regular set of these constants of motion.
ABSTRACT. This paper numerically computes the topological and smooth invariants of Eschenburg spaces with small fourth cohomology group, following Kruggel's determination of the Kreck-Stolz invariants of Eschenburg spaces that satisfy condition C. The GNU GMP arbitrary-precision library is utilised.
ABSTRACT. This note constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k-dimensional torus bundles over an l-dimensional torus. A central role is played by the Lax representation of a Bogoyavlenskij-Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.
ABSTRACT. Let M be a smooth compact oriented manifold without boundary, imbedded in a euclidean space E and let f be a smooth map of M into a Riemannian manifold N. An unknown state x in M is observed via X=x+su where s>0 is a small parameter and u is a white Gaussian noise. For a given smooth prior on M and smooth estimators g of the map f we have derived a second-order asymptotic expansion for the related Bayesian risk (see arXiv:0705.2540). In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilised in our earlier paper are naturally applicable to these regression problems.
ABSTRACT. 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 are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.
ABSTRACT. This paper studies smooth obstructions to integrability and proves two main results. First, it is shown that if a smooth topological n-torus admits a real-analytically completely integrable convex hamiltonian on its cotangent bundle, then the torus is diffeomorphic to the standard n-torus. This is the first known result where the smooth structure of a manifold obstructs complete integrability. Second, it is proven that each one of the Witten-Kreck-Stolz 7-manifolds admit a real-analytically completely integrable geodesic flow on its cotangent bundle. This gives examples of topological manifolds all of whose smooth structures admit a real-analytically completely integrable convex hamiltonian on its cotangent bundle. Additional examples are provided by Eschenburg and Aloff-Wallach spaces.
ABSTRACT. Let T be the nilpotent group of 4 × 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant Riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-Riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable.
ABSTRACT. Let Θ be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let γ be a smooth map from Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X=θ+∊ ξ where ∊ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior on Θ and smooth estimator g of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.
ABSTRACT. This note constructs a compact, real-analytic, Riemannian 4-manifold (Σ , g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) Σ is diffeomorphic to T2 × S2; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to real-analytic integrability beyond the topology of the configuration space.
ABSTRACT. --> If the geodesic flow of a compact Finslerian 3-manifold is completely integrable, and its singular set is a tame polyhedron, then the manifold's π1 is almost solvable (Butler 2005 Topology 44 769-89). Is this true for non-commutatively integrable geodesic flows? This paper constructs non-commutatively integrable optical Hamiltonians on the unit-sphere bundle of homogeneous spaces of PSL2R that have a real-analytic singular set. These flows are not tangent to a Lagrangian foliation with a tame singular set.
ABSTRACT. Let φt : M → M be a C1 flow and let L ⊂ M be open dense set with Γ = M-L. Assume L admits the structure of a Tk fibre bundle f : L → B. Say that φt is semisimple if Γ is a tamely embedded polyhedron and the fibres of f are φt-invariant. This paper shows that if (Σ, g) is a compact C2 finslerian 3-manifold whose geodesic flow on SΣ is semisimple with T3 fibres, then the fundamental group of Σ is almost 2-step polycyclic. Conversely, if Σ a compact 3-manifold with trivial 2nd homotopy group, then Σ admits a real-analytic Riemannian metric whose geodesic flow is semisimple with T3 fibres.
ABSTRACT. This paper studies the integrability of left-invariant geodesic flows on quotients of O+(2,1) × O+(2,1) × O+(2,1). It is proven that for each surface (M, h) of constant negative curvature, there is a Riemannian 8-manifold (Σ, k) such that Σ is homotopy equivalent to M and the geodesic flow of k is completely integrable. Amongst the manifolds Σ is an 8-dimensional manifold whose fundamental group is the free group on countably many generators. As a second consequence of this construction, the geodesic flow of a surface of constant negative curvative is embedded as a subsystem of an integrable geodesic flow. Secondly, let (M, h) be a compact geometric 3-manifold with the geometry of SL(2;R) or H2 × E1. Then there is a Riemannian 9-manifold (Σ, k) such that Σ is homotopy equivalent to M, the geodesic flow of k is completely integrable and the geodesic flow of h is a subsystem of k's. Combined with the results of Butler:1999, Bolsinov and Taimanov and Butler:2002, this shows that for each compact Seifert 3-manifold, there is a homotopy equivalent manifold that admits an integrable geodesic flow.
ABSTRACT. This paper studies completely integrable hamiltonian systems on T*M where M is a Tn+1 bundle over Tn with an R-split, free abelian monodromy group. For each periodic Toda lattice there is an integrable hamiltonian system on T*M with positive topological entropy. Bolsinov and Taimanov's example of an integrable geodesic flow with positive topological entropy fits into this general construction with the A(1)1 Toda lattice. Topological entropy is used to show that the flows associated to non-dual Toda lattices are typically topologically non-conjugate via an energy-preserving homeomorphism. The remaining cases are approached via the homology spectrum. An energy-preserving conjugacy implies the congruence of two rational quadratic forms over the unit group of a number field F. When F/Q is normal a classification of flows is obtained. In degree 3, this results from a well-known result of Gelfond; in higher degrees, the result is conditional on the conjecture that a rationally independent set of logarithms of algebraic numbers is algebraically independent over . Mathematics Subject Classification (2000) 58F17, 53D25, 37D40
ABSTRACT. Let g be a 2-step nilpotent Lie algebra; we say
g is non-integrable if, for a generic pair of points p,p'
∈ g*, the isotropy algebras do not commute:
[gp,gp'] ≠ 0.
Theorem: If G is a simply-connected 2-step nilpotent Lie group,
g = Lie(G) is non-integrable, D < G is a cocompact subgroup, and
g is a left-invariant Riemannian metric, then the geodesic
flow of
g on T*(D\G) is neither Liouville nor non-commutatively
integrable with C0 first integrals. The proof uses a
generalization of the rotation vector pioneered by Benardete and
Mitchell.
ABSTRACT. Let Tn denote the nilpotent group of n × n upper-triangular matrices with 1s on the diagonal. This note constructs left-invariant Riemannian and sub-Riemannian metrics on T4 ⊕ T3 whose geodesic flow has positive topological entropy.
ABSTRACT. (Joint work with Gabriel Paternain). We show that on each real form of so(4;C) there are Euler vector fields that have positive topological entropy. This result is used to obtain collective geodesic flows on homogeneous spaces with positive topological entropy. In this way, we obtain positive entropy geodesic flows on all compact rank-1 symmetric spaces (except S2, CP2 and OP2) and these geodesic flows commute with the standard "round" geodesic flows on these spaces.
ABSTRACT. This paper constructs a family of Liouville-integrable hamiltonian flows whose topologically generic trajectory is a translation on an invariant, embedded torus, but for which the probability of finding a trajectory whose Lyapunov exponents all vanish can be made arbitrarily close to zero. We also show that given any smooth symplectic diffeomorphism f : Σ2n → Σ2n of a compact symplectic manifold, we can construct a Liouville-integrable hamiltonian flow φt with a non-empty singular-point set C = K x Mf such that the Poincaré return map of φt | C equals idK × f.
ABSTRACT.This paper has four main results: (i) it shows that left-invariant geodesic flows on a broad class of 2-step nilmanifolds -- which are dubbed almost non-singular by Eberlein and Park & Lee -- are integrable in the non-commutative sense of Nehorosev; (ii) the left-invariant geodesic flows on all Heisenberg-Reiter nilmanifolds are Liouville integrable; (iii) the topological entropy of a left-invariant geodesic flow on a 2-step nilmanifold vanishes; (iv) there exist 2-step nilmanifolds with non-integrable left-invariant geodesic flows. It is also shown that for each of the integrable hamiltonians investigated here, there is a C2-open neighbourhood in C2(T* M) such that every integrable hamiltonian vector field in this neighbourhood must have wild first integrals.
ABSTRACT. The existence of a family of nilmanifolds which possess Riemannian metrics whose cogeodesic flow is Liouville-integrable is demonstrated. These homogeneous spaces are of the form D\H, where H is a connected, simply-connected and two-step nilpotent Lie group and D is a discrete, cocompact subgroup of H. The metric on these homogeneous spaces is obtained from a left-invariant metric on H. The topology of these nilmanifolds is quite rich; the first example of a Liouville-integrable geodesic flow on a manifold whose fundamental group possesses no commutative subgroup of finite index is obtained. It is shown that the conclusions of Taimanov's theorem do not obtain in the category of Liouville-integrable geodesic flows with smooth first integrals.
RÉSUMÉ. On construit une famille de nilvariétés qui possède une métrique Riemannienne avec un flot cogéodésique qui est intégrable [au sens de Liouville]. Ces espaces homogènes sont des quotients D\H où H est un groupe de Lie connexe et simplement connexe et D est un sousgroupe discrèt et cocompact de H. La métrique sur D\H vient d'une métrique sur H invariante à gauche. La topologie de ces espaces est très riche; en particulier, ils sont les premiers examples des variétés avec un groupe fondamentale non-presque-abelien qui possède un flot cogéodésique intégrable. On démontre que les conclusions du théorème de Taimanov, concernant la topologie d'un tel espace, sont fausses si les integrales du flot cogéodésique ne sont que C∞.
ABSTRACT. The existence of a family of nilmanifolds which possess Riemannian metrics with Liouville-integrable geodesic flow is demonstrated. These homogeneous spaces are of the form D\H, where H is a connected, simply-connected two-step nilpotent Lie group and D is a discrete, cocompact subgroup of H. The metric on D\H is obtained from a left-invariant metric on H. The topology of these nilmanifolds is quite rich; in particular, the first example of a Liouville-integrable geodesic flow on a manifold whose fundamental group possesses no commutative subgroup of finite index is obtained. It is shown that several of the conclusions of Taimanov's theorems on the topology of manifolds with real-analytically Liouville-integrable geodesic flows do not obtain in the smooth category. Additional properties of the geodesic flow are also demonstrated.
ABSTRACT. Recent examples of Liouville integrable geodesic flows on non-simply connected manifolds have shown that the topological implications of smooth Liouville integrability are dramatically different than the implications of real-analytic integrability. In particular a geodesic flow can be both smoothly integrable and have positive topological entropy. The examples due to Butler and Bolsinov and Taimanov are constructed from left-invariant metrics on Lie groups. In this paper, the degeneracy of the Poisson tensor on the dual algebra is shown to be the source of the large number of commuting first integrals, and additional examples of integrable geodesic flows are constructed.