scholarly journals Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kai Tao
Keyword(s):  
Lyapunov Exponent ◽  
Ergodic Theorem ◽  
Hölder Condition ◽  
Hölder Continuous ◽  
Subharmonic Functions ◽  
Birkhoff Ergodic Theorem ◽  
Special Case ◽  
Coupling Number

<p style='text-indent:20px;'>In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in <inline-formula><tex-math id="M1">\begin{document}$ E $\end{document}</tex-math></inline-formula> on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.</p>

Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycle with weak Liouville frequency

2013 ◽  
Vol 34 (4) ◽  
pp. 1395-1408 ◽  
Author(s):  
JIANGONG YOU ◽  
SHIWEN ZHANG
Keyword(s):  
Lyapunov Exponent ◽  
Density Of States ◽  
Large Deviation ◽  
Hölder Continuity ◽  
Integrated Density Of States ◽  
Diophantine Condition ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Subharmonic Functions ◽  
Large Deviation Theorem

AbstractFor analytic quasiperiodic Schrödinger cocycles, Goldshtein and Schlag [Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154 (2001), 155–203] proved that the Lyapunov exponent is Hölder continuous provided that the base frequency $\omega $ satisfies a strong Diophantine condition. In this paper, we give a refined large deviation theorem, which implies the Hölder continuity of the Lyapunov exponent for all Diophantine frequencies $\omega $, even for weak Liouville $\omega $, which improves the result of Goldshtein and Schlag.

scholarly journals Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Wafaa S. Sayed ◽  
Ahmed G. Radwan ◽  
Hossam A. H. Fahmy
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Degrees Of Freedom ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Financial Modeling ◽  
Design Flexibility ◽  
Special Case

The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.

The fractal dimension of quasi-periodic orbits

1998 ◽  
Vol 18 (6) ◽  
pp. 1467-1471
Author(s):  
BRYAN P. RYNNE
Keyword(s):  
Fractal Dimension ◽  
Asymptotic Behaviour ◽  
Periodic Orbits ◽  
Continuous Functions ◽  
Hölder Condition ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

Recently, Naito considered quasi-periodic orbits of Hölder continuous functions and obtained results relating the exponent in the Hölder condition to the asymptotic behaviour of the inclusion lengths of the $\epsilon$-almost periods of these orbits, and also to the fractal dimension of these orbits. In this paper we improve these results.

Digital Noise Generator Based on Bernoulli Chaotic Map

2012 ◽  
Vol 152-154 ◽  
pp. 1869-1873 ◽  
Author(s):  
Ricardo Francisco Martínez-González ◽  
José Alejandro Díaz-Méndez ◽  
César Enrique Rojas-López ◽  
Rubén Vázquez-Medina
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Ergodic Theorem ◽  
Chaotic Map ◽  
Digital Circuit ◽  
Noise Generator ◽  
Statistical Distributions ◽  
Digital Implementation ◽  
Operation Conditions

This paper describes the noise emulation in materials through the digital implementation of a chaotic noise generator based on the Bernoulli map. The bifurcation diagram, Lyapunov exponent and the Ergodic Theorem were used in order to determine the operation conditions under which the Bernoulli map is able to generate noise with different statistical distributions. A 32-bits digital circuit was designed and implemented in a FPGA, and it can be used to emulate and analyze the noise in different materials. The obtained results are consistent with the generalized Bernoulli map behavior.

scholarly journals A Note on the Birkhoff Ergodic Theorem

2017 ◽  
Vol 72 (1-2) ◽  
pp. 715-730
Author(s):  
Nikola Sandrić
Keyword(s):  
Ergodic Theorem ◽  
Birkhoff Ergodic Theorem

scholarly journals Extending the Applicability and Convergence Domain of a Fifth-Order Iterative Scheme under Hölder Continuous Derivative in Banach Spaces

2021 ◽  
pp. 258-270
Author(s):  
Debasis Sharma ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda
Keyword(s):  
Significant Contribution ◽  
Iterative Scheme ◽  
Local Analysis ◽  
Continuous Derivative ◽  
Hölder Condition ◽  
Convergence Domain ◽  
Hölder Continuous ◽  
Lipschitz Constants ◽  
Fifth Order ◽  
Extra Assumption

The most significant contribution made by this study is that the applicability and convergence domain of a fifth-order convergent nonlinear equation solver is extended. We use Hölder condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. This study generalizes the local analysis based on Lipschitz constants. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the involved operator. Finally, numerical experiments ensure that our analysis expands the utility of the considered scheme. In addition, the proposed technique produces a larger convergence domain, in comparison to the earlier study, without using any extra conditions.

Sign in / Sign up

Export Citation Format

Share Document