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Michał Rams  One of the best experts on this subject based on the ideXlab platform.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step SkewProducts and Elliptic Cocycles
Communications in Mathematical Physics, 2019CoAuthors: Lorenzo J. Díaz, K. Gelfert, Michał RamsAbstract:We study the fiber Lyapunov Exponents of step skewproduct maps over a complete shift of N , $${N\ge2}$$ N ≥ 2 , symbols and with C ^1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and Zero Exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ 2 × 2 matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ SL ( 2 , R ) cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov Exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given Exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov Exponent. The level set of the Zero Exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skewproducts arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov Exponent.

Nonhyperbolic step skewproducts: Entropy spectrum of Lyapunov Exponents
arXiv: Dynamical Systems, 2016CoAuthors: Lorenzo J. Díaz, Katrin Gelfert, Michał RamsAbstract:We study the fiber Lyapunov Exponents of step skewproduct maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and Zero Exponents. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov Exponent. The results are formulated in terms of LegendreFenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with given Exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov Exponent. The level set of Zero Exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov Exponent.
Lorenzo J. Díaz  One of the best experts on this subject based on the ideXlab platform.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step SkewProducts and Elliptic Cocycles
Communications in Mathematical Physics, 2019CoAuthors: Lorenzo J. Díaz, K. Gelfert, Michał RamsAbstract:We study the fiber Lyapunov Exponents of step skewproduct maps over a complete shift of N , $${N\ge2}$$ N ≥ 2 , symbols and with C ^1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and Zero Exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ 2 × 2 matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ SL ( 2 , R ) cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov Exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given Exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov Exponent. The level set of the Zero Exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skewproducts arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov Exponent.

Nonhyperbolic step skewproducts: Entropy spectrum of Lyapunov Exponents
arXiv: Dynamical Systems, 2016CoAuthors: Lorenzo J. Díaz, Katrin Gelfert, Michał RamsAbstract:We study the fiber Lyapunov Exponents of step skewproduct maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and Zero Exponents. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov Exponent. The results are formulated in terms of LegendreFenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with given Exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov Exponent. The level set of Zero Exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov Exponent.
K. Gelfert  One of the best experts on this subject based on the ideXlab platform.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step SkewProducts and Elliptic Cocycles
Communications in Mathematical Physics, 2019CoAuthors: Lorenzo J. Díaz, K. Gelfert, Michał RamsAbstract:We study the fiber Lyapunov Exponents of step skewproduct maps over a complete shift of N , $${N\ge2}$$ N ≥ 2 , symbols and with C ^1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and Zero Exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ 2 × 2 matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ SL ( 2 , R ) cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov Exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given Exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov Exponent. The level set of the Zero Exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skewproducts arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov Exponent.
Miguel A F Sanjuán  One of the best experts on this subject based on the ideXlab platform.

Local predictability and nonhyperbolicity through finite Lyapunov Exponent distributions in twodegreesoffreedom Hamiltonian systems.
Physical review. E Statistical nonlinear and soft matter physics, 2008CoAuthors: Juan C Vallejo, Ricardo L Viana, Miguel A F SanjuánAbstract:By using finite Lyapunov Exponent distributions, we get insight into both the local and global properties of a dynamical flow, including its nonhyperbolic behavior. Several distributions of finite Lyapunov Exponents have been computed in two prototypical fourdimensional phasespace Hamiltonian systems. They have been computed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals. We analyze how such distributions serve or not for tracing the orbit nature and local flow properties such as the unstable dimension variability, as the axes are allowed or not to tend to the largest stretching direction. The relationship between the largest and closest to Zero Exponent distribution is analyzed. It shows a linear dependency at short intervals, related to the number of degrees of freedom of the system. Finally, the hyperbolicity indexes, associated to the shadowing times, are calculated. They provide interesting information at very local scales, even when there are no Gaussian distributions and the values cannot be regarded as random variables.

Local predictability and nonhyperbolicity through finite Lyapunov Exponent distributions in twodegreesoffreedom Hamiltonian systems.
Physical Review E, 2008CoAuthors: Juan C Vallejo, Ricardo L Viana, Miguel A F SanjuánAbstract:Received 13 September 2007; revised manuscript received 5 November 2008; published 4 December 2008By using ﬁnite Lyapunov Exponent distributions, we get insight into both the local and global properties ofa dynamical ﬂow, including its nonhyperbolic behavior. Several distributions of ﬁnite Lyapunov Exponentshave been computed in two prototypical fourdimensional phasespace Hamiltonian systems. They have beencomputed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals. Weanalyze how such distributions serve or not for tracing the orbit nature and local ﬂow properties such as theunstable dimension variability, as the axes are allowed or not to tend to the largest stretching direction. Therelationship between the largest and closest to Zero Exponent distribution is analyzed. It shows a linear dependency at short intervals, related to the number of degrees of freedom of the system. Finally, the hyperbolicityindexes, associated to the shadowing times, are calculated. They provide interesting information at very localscales, even when there are no Gaussian distributions and the values cannot be regarded as random variables.DOI: 10.1103/PhysRevE.78.066204 PACS number s : 05.45. a
Esther Levenson  One of the best experts on this subject based on the ideXlab platform.

teachers knowledge of the nature of definitions the case of the Zero Exponent
The Journal of Mathematical Behavior, 2012CoAuthors: Esther LevensonAbstract:Abstract This paper focuses on three junior high school mathematics teachers and their knowledge of the nature of definitions. The mathematical context of Exponentiation is used as a springboard for discussing two aspects of definitions: their corresponding domains and the distinction and relationships between definitions, proofs, and theorems. Through interviews it was shown that some teachers are not aware that definitions and domains are intrinsically connected and some teachers believe that definitions may be proved. Findings also indicate that knowledge of the nature of definitions may be dependent on the context.