scholarly journals Quasianalytic monogenic solutions of a cohomological equation

2003 ◽  
Vol 164 (780) ◽  
pp. 0-0 ◽  
Author(s):  
S. Marmi ◽  
D. Sauzin
Keyword(s):  
Cohomological Equation

scholarly journals The cohomological equation and invariant distributions for horocycle maps

2012 ◽  
Vol 34 (1) ◽  
pp. 299-340 ◽  
Author(s):  
JAMES TANIS
Keyword(s):  
Compact Manifolds ◽  
Invariant Distributions ◽  
Cohomological Equation ◽  
Sobolev Estimates

AbstractWe study the invariant distributions for horocycle maps on $\Gamma \backslash SL(2, \mathbb {R})$and prove Sobolev estimates for the cohomological equation of horocycle maps. As an application, we obtain a rate of equidistribution for horocycle maps on compact manifolds.

scholarly journals A numerical approach based on $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials for solving fractional-order cohomological equations with variable coefficients

2021 ◽  
Author(s):  
Akbar Dehghan Nezhad ◽  
Mina Moghaddam Zeabadi
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Legendre Polynomials ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Cohomological Equation ◽  
Liouville Derivative ◽  
Cohomological Equations

This research presents a numerical approach to obtain the approximate solution of the n-dimensional cohomological equations of fractional order in continuous-time dynamical systems. For this purpose, the $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials (or n-DFMLPs) are introduced. The operational matrix of the fractional Riemann-Liouville derivative is constructed by employing n-DFMLPs. Our method transforms the cohomological equation of fractional order into a system of algebraic equations. Therefore, the solution of that system of algebraic equations is the solution of the associated cohomological equation. The error bound and convergence analysis of the applied method under the $ L^{2} $-norm is discussed. Some examples are considered and discussed to confirm the efficiency and accuracy of our method.

scholarly journals Central limit theorem for generalized Weierstrass functions

2019 ◽  
Vol 19 (01) ◽  
pp. 1950002
Author(s):  
Amanda de Lima ◽  
Daniel Smania
Keyword(s):  
Limit Theorem ◽  
Bounded Solution ◽  
Positive Lebesgue Measure ◽  
Lipschitz Continuous Function ◽  
Absolutely Continuous ◽  
Lipschitz Continuous ◽  
Expanding Map ◽  
Cohomological Equation ◽  
Weierstrass Functions ◽  
Absolutely Continuous Invariant

Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

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