The Experts below are selected from a list of 6498 Experts worldwide ranked by ideXlab platform
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 × 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 $C^2$ 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.
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 × 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.
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.
Benoît Loridant - One of the best experts on this subject based on the ideXlab platform.
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Fundamental group of tiles associated to quadratic canonical number systems
Mathematica Slovaca, 2008Co-Authors: Benoît LoridantAbstract:If A is a 2 × 2 expanding matrix with integral coefficients, and Open image in new window ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + Open image in new window is a self-affine plane tile of ℝ2, provided that its Two-Dimensional Lebesgue Measure is positive.
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Fundamental group of tiles associated to quadratic canonical number systems
Mathematica Slovaca, 2008Co-Authors: Benoît LoridantAbstract:If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ^2 a complete set of coset representatives of ℤ^2/ A ℤ^2 with |det( A )| elements, then the set ℐ defined by A ℐ = ℐ + is a self-affine plane tile of ℝ^2, provided that its Two-Dimensional Lebesgue Measure is positive. It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable. To a quadratic polynomial x ^2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B , one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2 A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2 A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.
Xin-han Dong - One of the best experts on this subject based on the ideXlab platform.
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STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS
Bulletin of the Australian Mathematical Society, 2020Co-Authors: Peng-fei Zhang, Xin-han DongAbstract:Abstract For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised Two-Dimensional Lebesgue Measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .
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Starlikeness and convexity of Cauchy transform on equilateral triangle
Complex Variables and Elliptic Equations, 2019Co-Authors: Peng-fei Zhang, Xin-han DongAbstract:Let Δ be an equilateral triangle, we define the Cauchy transform of the normalized Two-Dimensional Lebesgue Measure μ on Δ by F ( z ) = ∫ Δ d μ ( w ) z − w . We prove that F is univalent and starli...
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The Starlikeness of Cauchy Transform on Square
Computational Methods and Function Theory, 2019Co-Authors: Xin-han Dong, Ye Wang, Peng-fei ZhangAbstract:Suppose that K is the square with vertexes $$\{1, i, -1, -i\}$$ and $$\mu =\frac{1}{2}{\mathcal {L}}^2$$ is the normalized Two-Dimensional Lebesgue Measure on K, let F(z) be the Cauchy transform of $$\mu $$ . Dong et al. (Trans Am Math Soc 369:4817–4842, 2017) proved that F(z) is univalent in $$\widehat{\mathbb {C}} {\setminus } K$$ . In this paper, we show that F(z) is starlike in $$\widehat{\mathbb {C}} {\setminus } K$$ , but not convex in $$\widehat{\mathbb {C}} {\setminus } K$$ .
