Committee Feed
https://feeds.library.caltech.edu/people/Katok-A/committee.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:29:21 +0000I. Idempotent Multipliers of H¹ on the Circle. II. A Mean Oscillation Inequality for Rearrangements
https://resolver.caltech.edu/CaltechTHESIS:01222019-095854306
Authors: {'items': [{'id': 'Klemes-Ivo', 'name': {'family': 'Klemes', 'given': 'Ivo'}, 'show_email': 'NO'}]}
Year: 1985
DOI: 10.7907/5vae-8q84
<p>H<sup>1</sup>(T) is the space of integrable functions f on the circle T such that the Fourier coefficients f̂(n) vanish for negative integers n. A multiplier is by definition a map m of H<sup>1</sup> to itself such that the Fourier transform diagonal izes m. Let m̂(n) denote the diagonal coefficients of m for nonnegative n. Then m is called idempotent if each coefficient is zero or one.</p>
<p>Theorem: If m is idempotent, then the set of n for which m̂(n) = 1 is a finite Boolean combination of sets of nonnegative integers of the following three types: finite sets, arithmetic sequences, and lacunary sequences.</p>
<p>By definition, a sequence is lacunary if there is a real number q > 1 such that each term of the sequence is at least as large as q times the preceding term. The theorem implies a classification of the projections in H<sup>1</sup> which commute with translations, or, what is equivalent on the circle (but not on the line), of the closed, translation invariant subspaces which are complemented in H<sup>1</sup>. In the course of the proof, a 1 ower bound is obtai ned on the operator norm of a multiplier whose coefficients are 0 or greater than 1 in magnitude. This bound implies that the number of nonzero coefficients in disjoint intervals of the same length is the same, up to some factor depending on the norm of m, provided that both intervals are shorter than their distance from 0.</p>
<p>Part II is unrelated to Part I. There it is proved that a general expression measuring the oscillation of a function on an interval is minimized by the decreasing rearrangement of the function. A special case of this expression is the BMO norm for functions of bounded mean oscillation.</p>https://thesis.library.caltech.edu/id/eprint/11349Doublewell Tunneling via the Feynman-Kac Formula
https://resolver.caltech.edu/CaltechETD:etd-09062005-152643
Authors: {'items': [{'email': 'askell@ma.is', 'id': 'Hardarson-Askell', 'name': {'family': 'Hardarson', 'given': 'Askell'}, 'show_email': 'NO'}]}
Year: 1988
DOI: 10.7907/34ks-xy63
<p>We discuss asymptotics of the heat kernel [equation; see abstract in scanned thesis for details] and its x-derivatives when T, λ → ∞ and (T/λ) → 0 where H(λ) = - ((Δ/2) + λ²V) and where V is a double well. When the groundstate is localized in both wells for λ large we derive, by the Feynman-Kac formula, W.K.B. expansions of the groundstate, the first excited state and their gradients.</p>
<p>As a consequence we get a general asymptotic formula for the splitting of the two lowest eigenvalues, E₀(λ) and E₁(λ).</p>
<p>This formula allows us, in principle, always to go beyond the leading order given by [equation; see abstract in scanned thesis for details] where C is the action of a classical instanton.</p>
https://thesis.library.caltech.edu/id/eprint/3352Regularity of the Anosov Splitting and A New Description of the Margulis Measure
https://resolver.caltech.edu/CaltechETD:etd-05232007-105707
Authors: {'items': [{'id': 'Hasselblatt-Boris', 'name': {'family': 'Hasselblatt', 'given': 'Boris'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/qkw4-xf63
<p>The Anosov splitting into stable and unstable manifolds of hyperbolic dynamical systems has been known to be Holder continuous always and differentiable under bunching or dimensionality conditions. It has been known, by virtue of a single example, that it is not always differentiable. High smoothness implies some rigidity in several settings.</p>
<p>In this work we show that the right bunching conditions can guarantee regularity of the Anosov splitting up to being differentiable with derivative of Holder exponent arbitrarily close to one. On the other hand we show that the bunching condition used is optimal. Instead of providing isolated examples we prove genericity of the low-regularity situation in the absence of bunching. This is the first time a local construction of low-regularity examples is provided.</p>
<p>Based on this technique we indicate how horospheric foliations of nonconstantly curved symmetric spaces can be made to be nondifferentiable by a smoothly small perturbation.</p>
<p>In the last chapter the Hamenstadt-description of the Margulis measure is rendered for Anosov flows and with a simplified argument. The Margulis measure arises as a Hausdorff measure for a natural distance on (un)stable leaves that is adapted to the dynamics.</p>https://thesis.library.caltech.edu/id/eprint/1986Geodesic Flows on Manifolds of Negative Curvature with Smooth Horospheric Foliations
https://resolver.caltech.edu/CaltechETD:etd-05232007-115904
Authors: {'items': [{'id': 'Feres-Renato', 'name': {'family': 'Feres', 'given': 'Renato'}, 'show_email': 'NO'}]}
Year: 1989
DOI: 10.7907/f6yt-bf73
<p>We improve a result due to M. Kanai on the rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable (horospheric) foliation is smooth. More precisely, the main result proven here is: Let M be a closed C<sup>∞</sup> Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φ<sub>t</sub>: V → V on the unit tangent bundle V of M is C<sup>∞</sup>. Assume moreover that either (a) the sectional curvature of M satisfies -4 < K ≤ -1 or (b) the dimension of M is odd. Then the geodesic flow of M is C<sup>∞</sup>-isomorphic (i. e., conjugate under a C<sup>∞</sup> diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature.</p>https://thesis.library.caltech.edu/id/eprint/1988