An official website of the United States government
The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap distributions that are piecewise real analytic. Beyond that, however, very little is currently known about the general behavior of the slope gap distribution, including the number of points of non-analyticity or the tail. We show that the limiting gap distribution of slopes of saddle connections on a Veech translation surface is always piecewise real-analytic with finitely many points of non-analyticity. We do so by taking an explicit parameterization of a Poincaré section to the horocycle flow on SL(2,R)/SL(X,ω) associated to an arbitrary Veech surface SL(X,ω) and establishing a key finiteness result for the first return map under this flow. We use the finiteness result to show that the tail of the slope gap distribution of Veech surfaces always has quadratic decay.
more »
« less
Marked length rigidity for Fuchsian buildings
We consider finite $$2$$ -complexes $$X$$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT( $-1$ ) metrics on $$X$$ , which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $$X$$ . As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $$X$$ .
more »
« less
- PAR ID:
- 10133348
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 39
- Issue:
- 12
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 3262 to 3291
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Mulzer, Wolfgang; Phillips, Jeff M (Ed.)Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.) In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics.more » « less
-
We introduce a hardware–software system for rapidly characterizing liquid microjets for x-ray diffraction experiments. An open-source python-based software package allows for programmatic and automated data collection and analysis. We show how jet speed, length, and diameter are influenced by nozzle geometry, gas flow rate, liquid viscosity, and liquid flow rate. We introduce “jet instability” and “jet probability” metrics to help quantify the suitability of a given nozzle for x-ray diffraction experiments. Among our observations were pronounced improvements in jet stability and reliability when using asymmetric needle-tipped nozzles, which allowed for the production of microjects smaller than 250 nm in diameter, traveling faster than 120 m/s.more » « less
-
null (Ed.)Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric inducesa semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p . We also propose a conjectural generalization of this result for relativetwisted Kähler–Einstein metrics. Then we show that our conjecture holds trueif the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).more » « less
-
Dynamic Spectrum Access (DSA) radios typically select their radio channels according to their data networking goals, a defined DSA spectrum operating policy, and the state of the RF spectrum. RF spectrum sensing can be used to collect information about the state of the RF spectrum and prioritize which channels should be assigned for DSA radio waveform transmission and reception. This paper describes a Greedy Channel Ranking Algorithm (GCRA) used to calculate and rank RF interference metrics for observed DSA radio channels. The channel rankings can then be used to select and/or avoid channels in order to attain a desired DSA radio performance level. Experimental measurements are collected using our custom software-defined radio (SDR) system to quantify the performance of using GCRA for a DSA radio application. Analysis of these results show that both pre and post-detection average interference power metrics are the most accurate metrics for selecting groups of radio channels to solve constrained channel assignment problems in occupied gray space spectrum.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2018.12
