The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Víctor Jiménez López - One of the best experts on this subject based on the ideXlab platform.
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On the Lebesgue Measure of Li-Yorke pairs for interval maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract: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.
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On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract:We investigate the prevalence of Li-Yorke pairs for C2 and C3 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.
Daniel Lenz - One of the best experts on this subject based on the ideXlab platform.
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Singular spectrum of Lebesgue Measure zero for one-dimensional quasicrystals
Communications in Mathematical Physics, 2002Co-Authors: Daniel LenzAbstract:The spectrum of one-dimensional discrete Schr\"odinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue Measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples as e.g. the Rudin-Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.Comment: 14 page
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singular spectrum of Lebesgue Measure zero for one dimensional quasicrystals
Communications in Mathematical Physics, 2002Co-Authors: Daniel LenzAbstract:The spectrum of one-dimensional discrete Schrodinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue Measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.
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singular spectrum of Lebesgue Measure zero for one dimensional quasicrystals
arXiv: Mathematical Physics, 2001Co-Authors: Daniel LenzAbstract:The spectrum of one-dimensional discrete Schr\"odinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue Measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples as e.g. the Rudin-Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.
Henk Bruin - One of the best experts on this subject based on the ideXlab platform.
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On the Lebesgue Measure of Li-Yorke pairs for interval maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract: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.
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On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps
Communications in Mathematical Physics, 2010Co-Authors: Henk Bruin, Víctor Jiménez LópezAbstract:We investigate the prevalence of Li-Yorke pairs for C2 and C3 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.
Alexander Soshnikov - One of the best experts on this subject based on the ideXlab platform.
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Splitting of the low Landau levels into a set of positive Lebesgue Measure under small periodic perturbations
Communications in Mathematical Physics, 1997Co-Authors: Efim Dinaburg, Yakov G. Sinai, Alexander SoshnikovAbstract:We study the spectral properties of a two-dimensional Schrodinger operator with a uniform magnetic field and a small external periodic field: $$$$ where $$$$ and \(\), \(\) are small parameters. Representing \(\) as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume \(\) to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue Measure of the low bands is positive and proportional in the main order to \(\).
Tobias Kaiser - One of the best experts on this subject based on the ideXlab platform.
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Lebesgue Measure and integration theory on non archimedean real closed fields with archimedean value group
Proceedings of The London Mathematical Society, 2018Co-Authors: Tobias KaiserAbstract:Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue Measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the Measure and integration is obtained in a controlled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.
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Lebesgue Measure and integration theory on non‐archimedean real closed fields with archimedean value group
Proceedings of the London Mathematical Society, 2017Co-Authors: Tobias KaiserAbstract:Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue Measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry and valuation theory. We set up the construction in such a way that it is determined by a section of the valuation. If the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism. The range of the Measure and integration is obtained in a controled and tame way from the real closed field we start with. The main example is given by the case of the field of Puiseux series where the range is the polynomial ring in one variable over this field.Comment: Final version; to appear at the Proceedings of the London Mathematical Societ
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Lebesgue Measure theory and integration theory on arbitrary real closed fields
arXiv: Logic, 2014Co-Authors: Tobias KaiserAbstract:We establish for the category of semialgebraic sets and functions on arbitrary real closed fields a full Lebesgue Measure and integration theory such that the main results from the classical setting hold. The construction involves methods from model theory, o-minimal geometry, valuation theory and the theory of ordered abelian groups. We set up the construction in such a way that it is uniquely determined by data that can be formulated completely in terms of the given real closed field. We apply our integration theory to questions on semialgebraic geometry and analysis in the non-standard setting and also to questions on parameterized integrals on the reals.
