# Browsing by Subject "Torus (Geometry)"

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Item Constructions of open book decompositions(2007) Van Horn-Morris, Jeremy, 1978-; Gompf, Robert E., 1957-Show more We introduce the naive notion of a relative open book decomposition for contact 3-manifolds with torus boundary. We then use this to construct nice, minimal genus open book decompositions compatible with all of the universally tight contact structures (as well as a few others) on torus-bundles over S¹, following Honda's classification. In an accurate sense, we find Stein fillings of 'half' of the torus bundles. In addition, these give the first examples of open books compatible with the universally tight contact structures on circle bundles over higher genus surfaces, as well, following a pattern introduced by a branched covering of B⁴. Some interesting examples of open books without positive monodromy are emphasized.Show more Item Numerical studies of the standard nontwist map and a renormalization group framework for breakup of invariant tori(2004) Apte, Amit Shriram; Morrison, Philip J.Show more This thesis presents numerical explorations of area-preserving nontwist maps, and a renormalization group framework for the destruction of invariant tori. We study the phenomena of bifurcation and reconnection, and the emergence of meandering tori which are non-KAM invariant curves. We also study the breakup of shearless invariant tori with noble winding numbers using improved numerical techniques to implement Greene’s residue criterion. We interpret the breakup of invariant tori within a renormalization group framework by constructing renormalization group operators for the tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. We introduce coordinate transformations on the space of maps to relate these fixed points to each other. These transformations induce conjugacies between the corresponding operators, and provide a new perspective on the space of area-preserving maps.Show more Item Periodic orbit bifurcations and breakup of shearless invariant tori in nontwist systems(2006) Fuchss, Kathrin; Morrison, Philip J.Show more This thesis explores two nontwist systems: the spherical pendulum as an example of a continuous one and the standard nontwist map (SNM) as an example of a discrete one. Whereas the spherical pendulum is a concrete example of a physical system exhibiting nontwist phenomena, the SNM is an abstract, numerically easily accessible model permitting systematic studies of nontwist effects characteristic of a wide range of applications. For the spherical pendulum, a system that has captured physicists' and mathematicians' interest for centuries, the gradual progress in understanding this seemingly simple, but still not fully explored problem is outlined. The known solutions for the unforced (integrable) spherical pendulum are reviewed and approximated by power series. The approximations are then used to analytically calculate, for the vertically forced case, xed points and low-period periodic orbits. These are found to undergo collision phenomena typical for nontwist systems. For the SNM, a detailed cartography of parameter space is developed, based on periodic orbit collision curves and their branching thresholds, hyperbolic manifold reconnection thresholds, and the boundary for the onset of global chaos. This is used to nd meanders, multiple shearless curves, and extended scenarios for periodic orbit reconnection/collision. Based on Greene's residue criterion, the breakup of new types of shearless orbits: meanders, outer shearless tori, and a nonnoble torus is studied in detail within the framework of renormalization theory.Show more Item Renormalization, invariant tori, and periodic orbits for Hamiltonian flows(2001-05) Abad, Juan José, 1967-; Koch, Hans A.Show more Consideration is given to a family of renormalization transformations developed to study the existence of invariant tori in Hamiltonian systems. These transformations are used to construct invariant tori with self–similar frequency vectors, as well as sequences of periodic orbits approximating them, for near– integrable Hamiltonians in two or more degrees of freedom. Results on the location of the periodic orbits, and accumulation rates of these orbits to the invariant tori are presented. A numerical implementation of one of these transformations is used to to search for non–trivial fixed points in a reduced space of Hamiltonians with two degrees of freedom. Evidence supporting the existence of a fixed point that seems to be related to the break–up of golden invariant tori is provided. The critical indexand scaling found for this point are in good agreement with previous numerical experiments carried out for area preserving maps. Finally, a simple space of Hamiltonians in two degrees of freedom, all having a common invariant torus, is analyzed to gain insight vii into the dynamics of the flow on the invariant torus as the conjugacy to linear motion breaks down. By studying a particular one-parameter family of Hamiltonians, it is observed how the flow on the torus changes from being dense quasi–periodic to having an invariant quasi–periodic Cantor set. This transition is triggered by the appearance of a fixed point for the flow on the torus. Accumulation ratios of periodic orbits approaching the torus with this particular flow are found numerically, and shown to be different from those associated with known renormalization fixed pointsShow more