[Show abstract][Hide abstract] ABSTRACT: We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy.
[Show abstract][Hide abstract] ABSTRACT: Some Lyman continuum photons are likely to escape from most galaxies, and these can play an important role in ionizing gas around and between galaxies, including gas that gives rise to Lyman alpha absorption. Thus the gas surrounding galaxies and in the intergalactic medium will be exposed to varying amounts of ionizing radiation depending upon the distances, orientations, and luminosities of any nearby galaxies. The ionizing background can be recalculated at any point within a simulation by adding the flux from the galaxies to a uniform quasar contribution. Normal galaxies are found to almost always make some contribution to the ionizing background radiation at redshift zero, as seen by absorbers and at random points in space. Assuming that about 2 percent of ionizing photons escape from a galaxy like the Milky Way, we find that normal galaxies make a contribution of at least 30 to 40 percent of the assumed quasar background. Lyman alpha absorbers with a wide range of neutral column densities are found to be exposed to a wide range of ionization rates, although the distribution of photoionization rates for absorbers is found to be strongly peaked. On average, less highly ionized absorbers are found to arise farther from luminous galaxies, while local fluctuations in the ionization rate are seen around galaxies having a wide range of properties. Comment: 10 pages, 8 figures, references added, clarified explanation of first two equations
Monthly Notices of the Royal Astronomical Society 02/2003; 342(4). DOI:10.1046/j.1365-8711.2003.06625.x · 5.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We show that on a compact manifold of nonpositive curva-ture the volume of spheres (hence also that of balls) has an exact asymp-totic; it is purely exponential, and the growth rate equals the topological entropy. The resulting formula is the sharpest one which is known. It gener-alizes results of G.A. Margulis to the nonuniformly hyperbolic case. It improves the multiplicative asymptotic bound by G. Knieper.
[Show abstract][Hide abstract] ABSTRACT: We describe in detail a construction of weakly mixing C ∞ diffeo-morphisms preserving a smooth measure and a measurable Riemannian metric as well as Z k actions with similar properties. We construct those as a perturba-tion of elements of a nontrivial non-transitive circle action. Our construction works on all compact manifolds admitting a nontrivial circle action. It is shown in the appendix that a Riemannian metric preserved by a weakly mixing diffeomorphism can not be square integrable. 1. Relation between differentiable and measurable structure for smooth dynamical systems: a brief overview. Smooth ergodic theory studies mea-surable (or measure-theoretic, or ergodic) properties of differentiable dynamical systems with respect to natural invariant measures. (The word smooth will mean C ∞ unless explicitly stated otherwise). Such measures include smooth and, more generally, absolutely continuous measures such as Liouville measure for Hamilton-ian and Lagrangian systems or Haar measure for homogeneous systems, their limits such as SRB measures, invariant measures for uniquely ergodic systems, measures of maximal entropy on invariant locally maximal sets, and so on. There is a number of situations where a remarkable correspondence appears between the differentiable dynamical structure and properties of invariant measures. We will follow the gen-eral scheme of classifying representative behavior of smooth dynamical systems as elliptic, parabolic hyperbolic and partially hyperbolic elaborated in [HK]. One can divide positive results on interrelations between measurable and differ-entiable structures into two kinds which are not mutually exclusive: (i) Measurable structure determines differentiable structure completely or to a large extent (rigidity); (ii) measurable structure (and sometimes also topological or even differentiable structure) within certain classes of systems (such as perturbations of a given one) and on certain parts of phase space conforms to a certain set of standard models (stability). 1.1. Rigidity. Rigidity phenomena appear for systems with elliptic and parabolic behavior and for hyperbolic smooth actions of higher rank abelian groups.
Discrete and Continuous Dynamical Systems 12/1999; 6(1). DOI:10.3934/dcds.2000.6.61 · 0.83 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The full text of the dissertation is available as a Adobe Acrobat .pdf file (94 p.) ; Adobe Acrobat Reader required to view the file. Mode of access: World Wide Web. Thesis (Ph. D.)--Pennsylvania State University, 2002.