We prove an analogue of Klein combination theorem for Anosov subgroups by using a local-to-global principle for Morse quasigeodesics.
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.
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.
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.
In his 1985 paper Sullivan sketched a proof of his structural stability
theorem for group actions satisfying certain expansion-hyperbolicity axioms. In
this paper we relax Sullivan's axioms and introduce a notion of meandering
hyperbolicity for group actions on general metric spaces. This generalization
is substantial enough to encompass actions of certain non-hyperbolic groups,
such as actions of uniform lattices in semisimple Lie groups on flag manifolds.
At the same time, our notion is sufficiently robust and we prove that
meandering-hyperbolic actions are still structurally stable. We also prove some
basic results on meandering-hyperbolic actions and give other examples of such
In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2,C) to geometrically infinite discrete subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup Γ has a set of nonconical limit points with the cardinality of the continuum.
We prove a Morse lemma for regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings. We apply it to give a new coarse geometric characterization of Anosov subgroups of the isometry groups of such spaces simply as undistorted subgroups which are uniformly regular.
The reviewers contacted by the editors to evaluate this work have been unable to confirm that the main results are correct. Flaws that were identified by the reviewers in earlier versions of the paper have been addressed by the author. Although it is possible that future research will uncover a significant mistake in this paper or show that the conclusions are in error, I believe that publishing it may benefit the readership of the Journal and stimulate further work in mathematical physics on an important topic.
When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequencies of the corresponding eigenvectors. In this paper, we discuss this important problem further and propose a new method to organize those eigenvectors by defining and measuring 'natural' distances between eigenvectors using the Ramified Optimal Transport Theory followedby embedding them into a low-dimensional Euclidean domain. We demonstrate its effectiveness using a synthetic graph as well as a dendritic tree of a retinal ganglioncell of a mouse.