Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure

Energy Spectra, Altitude Profiles and Charge Ratios of Atmospheric Muons

We present a new measurement of air shower muons made during atmospheric ascent of the High Energy Antimatter Telescope balloon experiment. The muon charge ratio mu+ / mu- is presented as a function of atmospheric depth in the momentum interval 0.3-0.9 GeV/c. The differential mu- momentum spectra are presented between 0.3 and about 50 GeV/c at atmospheric depths between 13 and 960 g/cm^2. We compare our measurements with other recent data and with Monte Carlo calculations of the same type as those used in predicting atmospheric neutrino fluxes. We find that our measured mu- fluxes are smaller than the predictions by as much as 70% at shallow atmospheric depths, by about 20% at the depth of shower maximum, and are in good agreement with the predictions at greater depths. We explore the consequences of this on the question of atmospheric neutrino production.Comment: 11 pages, 8 figures, to appear in Phys. Rev. D (2000

Balloon Measurements of Cosmic Ray Muon Spectra in the Atmosphere along with those of Primary Protons and Helium Nuclei over Mid-Latitude

We report here the measurements of the energy spectra of atmospheric muons and of the cosmic ray primary proton and helium nuclei in a single experiment. These were carried out using the MASS superconducting spectrometer in a balloon flight experiment in 1991. The relevance of these results to the atmospheric neutrino anomaly is emphasized. In particular, this approach allows uncertainties caused by the level of solar modulation, the geomagnetic cut-off of the primaries and possible experimental systematics to be decoupled in the comparison of calculated fluxes of muons to measured muon fluxes. The muon observations cover the momentum and depth ranges of 0.3-40 GeV/c and 5-886 g/cmsquared, respectively. The proton and helium primary measurements cover the rigidity range from 3 to 100 GV, in which both the solar modulation and the geomagnetic cut-off affect the energy spectra at low energies.Comment: 31 pages, including 17 figures, simplified apparatus figure, to appear in Phys. Rev.