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This paper demonstrates sufficient conditions for the existence of KAM tori in a singly thermostated, integrable hamiltonian system with $n$ degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2--which include the Nos\'e thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of Legoll, Luskin & Moeckel, (Non-ergodicity of Nos\'e-Hoover dynamics, Nonlinearity, 22 (2009), pp. 1673--1694) to prove that a "typical" singly thermostated, integrable, real-analytic hamiltonian possesses a positive-measure set of invariant tori when the thermostat is weakly coupled. It also demonstrates a class of integrable hamiltonians, which, for a full-measure set of couplings, satisfies the same conclusion., Comment: 32 pages; 7 figures

This paper studies the dynamics of a family of hamiltonian systems that originate from Friedman-Lema\^itre-Robertson-Walker space-times with a coupled field and non-zero curvature. In four distinct cases, previously considered by Maciejewski, Przybylska, Stachowiak & Szydowski, it is shown that there are homoclinic connections to invariant submanifolds and the connections split. These results imply the non-existence of a real-analytic integral independent of the hamiltonian., Comment: 21 pages, 4 figures

This note studies 1 and 2 degree of freedom hamiltonian systems that are thermostated by a single-variable thermostat. Under certain conditions on the hamiltonian and thermostat, the existence of a horseshoe in the flow of the thermostated system is proven., Comment: 16 pages; 1 figure

This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM tori in a singly thermostated, 1 degree-of-freedom hamiltonian vector field. This result is applied to 4 important single thermostats in the literature and it is shown that in each case, if the hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends results of Legoll, Luskin & Moeckel., Comment: 27 pages, 8 figures

Let ({\Sigma}, g) be a compact $C^2$ finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then ${\pi}_1({\Sigma})$ is almost polycyclic. On the other hand, if {\Sigma} is a compact, irreducible 3-manifold and ${\pi}_1({\Sigma})$ is infinite polycyclic while ${\pi}_2({\Sigma})$ is trivial, then {\Sigma} admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven., Comment: 28 pages. Published in 2005

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2 \times S^2$ ; and (3) the limit set of the geodesic flow on the universal cover is dense. This shows there are obstructions to realanalytic integrability beyond the topology of the configuration space., Comment: 8 pages. Published in 2006

Let $H(q,p) = \frac12 | p |^2 + V(q)$ be an $n$-degree of freedom $C^r$ mechanical Hamiltonian on the cotangent bundle of the $n$-torus where $r>2n+2$. When the metric $| * |$ is flat, the Nos\'e-thermostated system associated to $H$ is shown to have a positive-measure set of invariant tori near the infinite temperature limit. This is shown to be true for all variable mass thermostats similar to Nos\'e's, too. These results complement results of Legoll, Luskin & Moeckel and the author., Comment: 11 pages

Let H(q,p) = p^2/2 + V(q) be a 1-degree of freedom mechanical Hamiltonian with a C^n periodic potential V where n>4. The Nos\'e-thermostated system associated to H is shown to have invariant tori near the infinite temperature limit. This is shown to be true for all thermostats similar to Nos\'e's. These results complement the result of Legoll, Luskin and Moeckel who proved the existence of such tori near the decoupling limit., Comment: 10 pages

Let $\Sigma$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$ upper triangular matrices with unity along the diagonal. The Lie algebra $t_4$ of $T_4$ has the standard basis $\{X_{ij}\}$ of matrices with $0$ everywhere but in the $(i,j)$ entry, which is unity. Let $g$ be the Carnot metric, a sub-riemannian metric, on $T_4$ for which $X_{i,i+1}$, $(i=1,2,3)$, is an orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow of $g$ is algebraically non-integrable. This note proves that the geodesic flow of that Carnot metric on $T \Sigma$ has positive topological entropy and is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich., Comment: 3 figures

Let S be a compact, connected surface and H in C^2(T^* S) a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Euler characteristic of S when H is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If H is 2-semisimple, then S has non-negative Euler characteristic; if H is 1-semisimple and reversible, then S has positive Euler characteristic., Comment: 7 pages, 1 figure; removed reversibility hypothesis and renamed in v2; to appear in DCDS-A

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians., Comment: 24 pages, 1 figure; v2 corrects typo in online abstract; v3 includes new title (was: A Weak Liouville-Arnold Theorem), re-arrangement of introduction, re-numbering of main theorems; v4 updates the authors' email and physical addresses, clarifies notation in section 4. Final version

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., Comment: 7 tables, 2 figures, C++ source code is available at http://www.maths.ed.ac.uk/~lbutler/kspace/, v2 has equation keys removed

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., Comment: 25 pages, 5 figures, v2 includes thanks

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., Comment: 46 pages, 9 tables, 3 figures. To appear in Math. Proc. Camb. Phi. Soc

Let T be the nilpotent group of 4 x 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., Comment: 1 figure, 1 table

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 Eschenburgh and Aloff-Wallach spaces., Comment: 19 pages; v2: Proposition 4.1 is corrected. Main results are unchanged

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., Comment: Final version to appear in CMP. Two new remarks have been added as well as some numerical calculations for metric entropy

Let \Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \gamma be a smooth map \Theta into a riemannian manifold \Lambda. An unknown state \theta \in \Theta is observed via X=\theta+\epsilon \xi where \epsilon>0 is a small parameter and \xi is a white Gaussian noise. For a given smooth prior on \Theta and smooth estimator g of the map \gamma we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \Theta and \Lambda, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \gamma is found based on the modern theory of harmonic maps and hypo-elliptic differential operators., Comment: 20 pages, no figures published version includes correction to eq.s 31, 41, 43