arXiv subjects

Philip Boyland

Publications and source records attributed to Philip Boyland.

28 records · Page 2Linked to original sources

arXiv2006

The Burau estimate for the entropy of a braid

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for $t=-1$. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved.

math.DS↗

arXiv2005

Relative Periodic Solutions of the Complex Ginzburg-Landau Equation

A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the one-dimensional complex Ginzburg-Landau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation's symmetry group. With the method used, relative periodic solutions are represented by a space-time Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories.

nlin.CD↗

arXiv2004

Semiconjugacies to angle-doubling

A simple consequence of a theorem of Franks says that whenever a continuous map, $g$, is homotopic to angle doubling on the circle it is semiconjugate to it. We show that when this semiconjugacy has one disconnected point inverse, then the typical point in the circle has a point inverse with uncountably many connected components. Further, in this case the topological entropy of $g$ is strictly larger than that of angle doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-$D$ circle maps with $D > 2$.

math.DS↗

arXiv2003

Dynamics of two-dimensional time-periodic Euler fluid flows

This paper investigates the dynamics of time-periodic Euler flows in multi-connected, planar fluid regions which are ``stirred'' by the moving boundaries. The classical Helmholtz theorem on the transport of vorticity implies that if the initial vorticity of such a flow is generic among real-valued functions in the $C^k$-topology ($k \geq 2$) or is $C^ω$ and nonconstant, then the flow has zero topological entropy. On the other hand, it is shown that for constant initial vorticity there are stirring protocols which always yield time-periodic Euler flows with positive entropy. These protocols are those that generate flow maps in pseudoAnosov isotopy classes. These classes are a basic ingredient of the Thurston-Nielsen theory and a further application of that theory shows that pseudoAnosov stirring protocols with generic initial vorticity always yield solutions to Euler's equations for which the sup norm of the gradient of the vorticity grows exponentially in time. In particular, such Euler flows are never time-periodic.

math.DS↗

arXiv1999

Topological fluid mechanics of point vortex motions

Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. The reduction of each system using its symmetries is described in detail. Restricting to three vortices with zero net circulation, each reduced system is described by a one degree of freedom Hamiltonian. The phase portrait of this reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston-Nielsen theory is then used to analyse these isotopy classes, and in certain cases strong conclusions about the dynamics of the advection can be made.

chao-dyn↗

arXiv1999

Isotopy Stability of Dynamics on Surfaces

This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relations is proved and necessary and sufficient conditions for the isotopy stability of an equivalence class are given. It is also shown that most the dynamics of the minimal representative persist under isotopy in the sense that any isotopic map has an invariant set that is semiconjugate to it.

math.DS↗

arXiv1996

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 geodesics on closed surfaces and hyperbolic manifolds.

math.DS↗

arXiv1996

Dynamical stability in Lagrangian systems

This paper surveys various results concerning stability for the dynamics of Lagrangian (or Hamiltonian) systems on compact manifolds. The main, positive results state, roughly, that if the configuration manifold carries a hyperbolic metric, \ie a metric of constant negative curvature, then the dynamics of the geodesic flow persists in the Euler-Lagrange flows of a large class of time-periodic Lagrangian systems. This class contains all time-periodic mechanical systems on such manifolds. Many of the results on Lagrangian systems also hold for twist maps on the cotangent bundle of hyperbolic manifolds. We also present a new stability result for autonomous Lagrangian systems on the two torus which shows, among other things, that there are minimizers of all rotation directions. However, in contrast to the previously known \cite{hedlund} case of just a metric, the result allows the possibility of gaps in the speed spectrum of minimizers. Our negative result is an example of an autonomous mechanical Lagrangian system on the two-torus in which this gap actually occurs. The same system also gives us an example of a Euler-Lagrange minimizer which is not a Jacobi minimizer on its energy level.

math.DS↗

arXiv1994

Dual billiards, twist maps, and impact oscillators

In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards.

math.DS↗

arXiv1992

Weak disks of Denjoy minimal sets

This paper investigates the existence of Denjoy minimal sets and, more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It is shown that for the full two-shift, the collection of such invariant sets with the weak topology contains topological balls of all finite dimensions. One implication is an analogous result that holds for diffeomorphisms with transverse homoclinic points. It is also shown that the union of Denjoy minimal sets is dense in the two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.

math.DS↗