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Geometric infiniteness in negatively pinched Hadamard manifolds

(2020)

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $\Gamma$ has a set of nonconical limit points with cardinality of continuum.

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A combination theorem for Anosov subgroups

(2019)

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We prove an analogue of Klein combination theorem for Anosov subgroups by using a local-to-global principle for Morse quasigeodesics.

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Cover page of Polymer stress growth in viscoelastic fluids in oscillating extensional flows with applications to micro-organism locomotion

Polymer stress growth in viscoelastic fluids in oscillating extensional flows with applications to micro-organism locomotion

(2019)

© 2019 Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers. We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill geometry. The stress from these simulations is compared with the theoretical calculation of stress in the decoupled (given flow) case, and similar stress behavior is observed. The flow around tips of a time-reversible flexing filament in a viscoelastic fluid is shown to exhibit an oscillating extension along particle trajectories, and the stress response exhibits similar transitions. However in the high amplitude, high De regime the stress feedback on the flow leads to non time-reversible particle trajectories that experience asymmetric stretching and compression, and the stress grows more significantly in this regime. These results help explain past observations of large stress concentration for large amplitude swimmers and non-monotonic dependence on De of swimming speeds.

Cover page of Noncoherence of some lattices in Isom(Hn)

Noncoherence of some lattices in Isom(Hn)

(2019)

We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.

Cover page of Multiple-scale analysis on the radiation within the coupled KdV
  equations

Multiple-scale analysis on the radiation within the coupled KdV equations

(2019)

A multiple scale model of the nonlinearly coupled KdV equations is established to predict mechanism of interaction of equatorial Rossby waves and barotropic waves in certain case. Analytically, predicted precursor radiation is a centrosymmetric object and is shown in excellent quantitative agreement with numerical simulations; furthermore, the multiple scale model elucidates the salient mechanisms of the interaction of solitary waves and the mechanism for radiation. While the atmosphere-ocean science community is very interested in theoretical studies of tropical wave interactions and in developing reduced dynamical models that can explain some key features of equatorial phenomena, our analytic predictions quantitively explain formation of radiation during interaction in Biello's model beyond qualitative level.

Ping-pong in Hadamard manifolds

(2019)

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when $f$ is hyperbolic.

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Cover page of Patterson-Sullivan theory for Anosov subgroups

Patterson-Sullivan theory for Anosov subgroups

(2019)

We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

Cover page of Sullivan's structural stability of expanding group actions

Sullivan's structural stability of expanding group actions

(2019)

In his 1985 paper Sullivan sketched a proof of his structural stability theorem for group actions satisfying certain expansion-hyperbolicity axioms. We generalize the theorem by weakening these axioms substantially, while adding more details to Sullivan's original proof. We then present a number of examples satisfying Sullivan's axioms, such as Anosov subgroups of Lie groups as well as hyperbolic and non-hyperbolic groups acting on metric spaces.

Cover page of Unified theory for finite Markov chains

Unified theory for finite Markov chains

(2019)

© 2019 We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded S and the original S. Analyzing the normal forms also provides an estimate on the mixing time.