On Some Holomorphic Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

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Cohomological equations and invariant distributions for minimal circle diffeomorphisms

Duke Mathematical Journal ◽

10.1215/00127094-1345662 ◽

2011 ◽

Vol 158 (3) ◽

pp. 501-536 ◽

Cited By ~ 8

Author(s):

Artur Avila ◽

Alejandro Kocsard

Keyword(s):

Invariant Distributions ◽

Circle Diffeomorphisms ◽

Cohomological Equations

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On the normal solvability of cohomological equations on compact topological spaces

Recent Progress in Operator Theory ◽

10.1007/978-3-0348-8793-9_4 ◽

1998 ◽

pp. 75-87 ◽

Cited By ~ 2

Author(s):

G. Belitskii ◽

Yu. I. Lyubich

Keyword(s):

Topological Spaces ◽

Normal Solvability ◽

Cohomological Equations

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ELLIPTIC OPERATORS AND SOLUTIONS OF COHOMOLOGICAL EQUATIONS FOR GEODESIC FLOWS WITH HYPERBOLIC BEHAVIOR

International Journal of Mathematics ◽

10.1142/s0129167x91000387 ◽

1991 ◽

Vol 02 (06) ◽

pp. 701-709

Author(s):

SVETLANA KATOK

Keyword(s):

Negative Curvature ◽

Infinitesimal Generator ◽

Geodesic Flows ◽

Constant Negative Curvature ◽

Smooth Functions ◽

Negatively Curved ◽

Finite Dimensional ◽

Unit Tangent Bundle ◽

Hyperbolic Behavior ◽

Cohomological Equations

In this paper we study the space of smooth functions on the unit tangent bundle SM to a compact negatively curved surface M that are eigenfunctions of the infinitesimal generator of the action of SO(2) on SM, and that have zero integrals over all periodic orbits of the geodesic flow on SM. It is proved that the space of such functions is finite dimensional. In the case of constant negative curvature a complete description of this space is obtained.

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ε-neighbourhoods of orbits of parabolic diffeomorphisms and cohomological equations

Nonlinearity ◽

10.1088/0951-7715/27/12/3005 ◽

2014 ◽

Vol 27 (12) ◽

pp. 3005-3029 ◽

Cited By ~ 3

Author(s):

Maja Resman

Keyword(s):

Cohomological Equations

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

10.22541/au.161564768.87107277/v1 ◽

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.

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