scholarly journals Power spectrum of the geodesic flow on hyperbolic manifolds

2015 ◽  
Vol 8 (4) ◽  
pp. 923-1000 ◽  
Author(s):  
Semyon Dyatlov ◽  
Frédéric Faure ◽  
Colin Guillarmou
Keyword(s):  
Power Spectrum ◽  
Geodesic Flow ◽  
Hyperbolic Manifolds

scholarly journals A note on Myrberg points and ergodicity

2005 ◽  
Vol 96 (1) ◽  
pp. 107 ◽  
Author(s):  
Kurt Falk
Keyword(s):  
Geodesic Flow ◽  
Kleinian Group ◽  
Hyperbolic Manifold ◽  
Strong Relationship ◽  
Hyperbolic Manifolds ◽  
Limit Set

The main purpose of this note is to further clarify the strong relationship between ergodicity and Myrberg type dynamics in hyperbolic manifolds. This is achieved mainly via a new suggestive proof of the fact that the Myrberg limit set of a Kleinian group is of full Patterson measure if the geodesic flow on the associated hyperbolic manifold is ergodic with respect to the Patterson-Sullivan measure.

scholarly journals Ruelle and Quantum Resonances for Open Hyperbolic Manifolds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1445-1480
Author(s):  
Charles Hadfield
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifolds ◽  
Quantum Resonances ◽  
Classical Quantum ◽  
Convex Cocompact ◽  
Quantum Correspondence

Abstract We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectra of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.

scholarly journals Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

2021 ◽  
Vol 17 (0) ◽  
pp. 401
Author(s):  
Dubi Kelmer ◽  
Hee Oh
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
General Theorem ◽  
Hyperbolic Manifolds ◽  
Quantitative Estimates ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Tubular Neighborhoods ◽  
Logarithm Laws ◽  
Metric Balls

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

scholarly journals Lagrangian systems on hyperbolic manifolds

1999 ◽  
Vol 19 (5) ◽  
pp. 1157-1173 ◽  
Author(s):  
PHILIP BOYLAND ◽  
CHRISTOPHE GOLÉ
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Invariant Sets ◽  
Lagrangian System ◽  
Mather Theory ◽  
Closed Surfaces ◽  
Bounded Distance ◽  
Time Periodic

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

The on-line use of histogram data for high-speed correction and alignment of electron microscope images

1984 ◽  
Vol 42 ◽  
pp. 556-557
Author(s):  
William Krakow
Keyword(s):  
Power Spectrum ◽  
High Speed ◽  
Direct Memory Access ◽  
Digital Television ◽  
Contrast Transfer Function ◽  
The Past ◽  
High Resolution Tem ◽  
On Line ◽  
Correction Of Astigmatism ◽  
The Fourier Transform

In the past few years on-line digital television frame store devices coupled to computers have been employed to attempt to measure the microscope parameters of defocus and astigmatism. The ultimate goal of such tasks is to fully adjust the operating parameters of the microscope and obtain an optimum image for viewing in terms of its information content. The initial approach to this problem, for high resolution TEM imaging, was to obtain the power spectrum from the Fourier transform of an image, find the contrast transfer function oscillation maxima, and subsequently correct the image. This technique requires a fast computer, a direct memory access device and even an array processor to accomplish these tasks on limited size arrays in a few seconds per image. It is not clear that the power spectrum could be used for more than defocus correction since the correction of astigmatism is a formidable problem of pattern recognition.

Log-log scale roughness spectroscopy of surfaces

1993 ◽  
Vol 51 ◽  
pp. 224-225
Author(s):  
P. Fraundorf ◽  
B. Armbruster
Keyword(s):  
Power Spectrum ◽  
Autocorrelation Function ◽  
Power Spectra ◽  
Scanning Force Microscopy ◽  
Scanning Tunneling ◽  
Tunneling Microscopy ◽  
Force Microscopy ◽  
Scanning Force ◽  
Lateral Structure ◽  
Scale Roughness

Optical interferometry, confocal light microscopy, stereopair scanning electron microscopy, scanning tunneling microscopy, and scanning force microscopy, can produce topographic images of surfaces on size scales reaching from centimeters to Angstroms. Second moment (height variance) statistics of surface topography can be very helpful in quantifying “visually suggested” differences from one surface to the next. The two most common methods for displaying this information are the Fourier power spectrum and its direct space transform, the autocorrelation function or interferogram. Unfortunately, for a surface exhibiting lateral structure over several orders of magnitude in size, both the power spectrum and the autocorrelation function will find most of the information they contain pressed into the plot’s origin. This suggests that we plot power in units of LOG(frequency)≡-LOG(period), but rather than add this logarithmic constraint as another element of abstraction to the analysis of power spectra, we further recommend a shift in paradigm.

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