Keywords:-
Keywords:
Surfaces of Section; Return maps
Article Content:-
Abstract
New 2-dimensional Birkhoff surfaces of sections are defined for the desymmetrized P SL(2, Z) group. The new object is demonstrated to be apt to study the geodesics flow solution of the Hamiltonian problem. The new definitions of the return maps of the new 2-dimensional Birkhoff surface of section are provided with; The demonstration relays on the self-adjointed-ness of the operators on which the conjugacy subclasses needed in the application to reduced surds act
References:-
References
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E. B. Bogomolny, M. Carioli, Quantum maps from transfer operators, Physica D: Nonlinear Phenomena 67, 88 (1993).
E. Brenner, F. Spinu, Artin formalism for Selberg zeta functions of co-finite Kleinian groups, Journal de th´eorie des nombres de Bordeaux 21, 59 (2009).
M. Eisele, D. Mayer, Dynamical zeta functions for Artins billiard and the Venkov-Zograf factorization formula, Physica D: Nonlinear Phenomena 70, 342 (1994).
P. Giulietti, On Transfer Operators for Anosov Flows, eprint http://www1.mat.uniroma1.it/PhD/TESI/ARCHIVIO/giuliettipaolo.p (2010).
F. Faure, M. Tsujii, Prequantum transfer operator for symplectic Anosov diffeomorphism, eprint arXiv:1206.0282.
T. Marty, Anosov flows and Birkhoff sections, eprint https://theses.hal.science/tel-03510071/document (2016).
M. Asaoka, C. Bonatti, T. Marty, Oriented Birkhoff sections of Anosov flows, eprint arXiv:2212.06483.
C. H. Chang, Quantization conditions in Bogomolny’s transfer operator method, Phys. Rev. E 66, 056202 (2002).
C. C. Tsang, Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity, eprint arXiv:2206.09586 (2006).
O. M. Lecian, More Suitable Definition of Quadratic Surds, The Open Mathematics, Statistics and Probability Journal 10, 8 (2020).
D. H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z), Bull. Amer. Math. Soc. 25, 55 (1991).
A. Fathi, F. Laudenbach, V. Po´enaru (Editors):Travaux de Thurston sur les surfaces-seminaire Orsay. Asterisques 66-67, (1979).
D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology 22, 299 (1983).
S. Goodman, Dehn surgery on Anosov flows, Geometric Dynamics, in Lecture Notes in Mathematics, 1007, 300 (J. Palis ed.), Springer Berlin Heidelberg, Chapel Hill- USA (1980).
M. Handel, W. Thurston, Anosov flows on new 3-manifolds, Inv. Math. 59, 95 (1980).
M. Gerber, A.Katok, Smooth models of Thurstons pseudo-Anosov maps, Annales scientifiques de l’E.N.S., 15, 173 (1982).
M. Ratner, Markov decomposition for an U-flow on a three-dimensional manifold. Math. Notes 6 (1969), 880-886.
A. B. Venkov, The Artin-Takagi formula for Selberg’s zeta-function and the Roelcke conjecture, Dokl. Akad. Nauk SSSR 247 (1979), 540-543; English transl. in Soviet Math. Dokl. 20 (1979).
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A. Katok, G. Knieper , M. Pollicott, H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. math. 98, 581-597 (1989).
A. Manning, Topological entropy for geodesic flows, Annals of Mathematics 110, 567(1979).
C. A. Morales, Sectional-Anosov flows, Monatshefte fuer Mathematik 159, 253 (2010).
S. Cantat, Bers and H´enon, Painlev´e and Schroedinger, Duke Math. J. 149, 411 (2009).
R. Phillips, P. Sarnak, Geodesics in homology classes, Duke Math. J. 55, 287 (1987).
G. A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature. Funct. Anal. Its Appl. 3, 335 (1969).
D. V. Anosov, Geodesicflows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) [MR 39 n3527].
J. F. Plante, Homology of Closed Orbits of Anosov Flows, Proceedings of the American Mathematical Society 37, 297(1973).
Sharp, R. (1993). Closed orbits in homology classes for Anosov flows. Ergodic Theory and Dynamical Systems, 13(2), 387-408.
B. Farb, D. Margalit, A primer on mapping class groups, Princeton Mathematical Series 49. Princeton University Press, Princeton- USA (2012).
Dyatlov, S., Faure, C. Guillarmou, Power spectrum of the geodesic flow on hyperbolic manifolds, Analysis and PDE 8, 923 (2015)
A. B. Venkov, P. G. Zograf, On analogues of the Artin factorization in the spectral theory of automorphic functions connected with the induced representations of Fuchsiam groups, Math. USSR Izv. 21, 435 (1983).
E. B. Bogomolny, M. Carioli, Quantum maps from transfer operators, Physica D: Nonlinear Phenomena 67, 88 (1993).
E. Brenner, F. Spinu, Artin formalism for Selberg zeta functions of co-finite Kleinian groups, Journal de th´eorie des nombres de Bordeaux 21, 59 (2009).
M. Eisele, D. Mayer, Dynamical zeta functions for Artins billiard and the Venkov-Zograf factorization formula, Physica D: Nonlinear Phenomena 70, 342 (1994).
P. Giulietti, On Transfer Operators for Anosov Flows, eprint http://www1.mat.uniroma1.it/PhD/TESI/ARCHIVIO/giuliettipaolo.p (2010).
F. Faure, M. Tsujii, Prequantum transfer operator for symplectic Anosov diffeomorphism, eprint arXiv:1206.0282.
T. Marty, Anosov flows and Birkhoff sections, eprint https://theses.hal.science/tel-03510071/document (2016).
M. Asaoka, C. Bonatti, T. Marty, Oriented Birkhoff sections of Anosov flows, eprint arXiv:2212.06483.
C. H. Chang, Quantization conditions in Bogomolny’s transfer operator method, Phys. Rev. E 66, 056202 (2002).
C. C. Tsang, Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity, eprint arXiv:2206.09586 (2006).
O. M. Lecian, More Suitable Definition of Quadratic Surds, The Open Mathematics, Statistics and Probability Journal 10, 8 (2020).
D. H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for PSL(2,Z), Bull. Amer. Math. Soc. 25, 55 (1991).
A. Fathi, F. Laudenbach, V. Po´enaru (Editors):Travaux de Thurston sur les surfaces-seminaire Orsay. Asterisques 66-67, (1979).
D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology 22, 299 (1983).
S. Goodman, Dehn surgery on Anosov flows, Geometric Dynamics, in Lecture Notes in Mathematics, 1007, 300 (J. Palis ed.), Springer Berlin Heidelberg, Chapel Hill- USA (1980).
M. Handel, W. Thurston, Anosov flows on new 3-manifolds, Inv. Math. 59, 95 (1980).
M. Gerber, A.Katok, Smooth models of Thurstons pseudo-Anosov maps, Annales scientifiques de l’E.N.S., 15, 173 (1982).
M. Ratner, Markov decomposition for an U-flow on a three-dimensional manifold. Math. Notes 6 (1969), 880-886.
A. B. Venkov, The Artin-Takagi formula for Selberg’s zeta-function and the Roelcke conjecture, Dokl. Akad. Nauk SSSR 247 (1979), 540-543; English transl. in Soviet Math. Dokl. 20 (1979).
W. Roelcke, Sitzungs. Akad. Wiss. Math. Kl., 1953-1955,.B.4, Abh. Heidelborg, 1956.
E.B. Bogomolny, Comm. At. Mol. Phys. 25 (1990) 67.
E.B. Bogomolny, Nonlinearity 5 (1992) 805.
A. Katok, G. Knieper , M. Pollicott, H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. math. 98, 581-597 (1989).
A. Manning, Topological entropy for geodesic flows, Annals of Mathematics 110, 567(1979).
C. A. Morales, Sectional-Anosov flows, Monatshefte fuer Mathematik 159, 253 (2010).
S. Cantat, Bers and H´enon, Painlev´e and Schroedinger, Duke Math. J. 149, 411 (2009).
R. Phillips, P. Sarnak, Geodesics in homology classes, Duke Math. J. 55, 287 (1987).
G. A. Margulis, Applications of ergodic theory to the investigation of manifolds of negative curvature. Funct. Anal. Its Appl. 3, 335 (1969).
D. V. Anosov, Geodesicflows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) [MR 39 n3527].
J. F. Plante, Homology of Closed Orbits of Anosov Flows, Proceedings of the American Mathematical Society 37, 297(1973).
Sharp, R. (1993). Closed orbits in homology classes for Anosov flows. Ergodic Theory and Dynamical Systems, 13(2), 387-408.
B. Farb, D. Margalit, A primer on mapping class groups, Princeton Mathematical Series 49. Princeton University Press, Princeton- USA (2012).
Dyatlov, S., Faure, C. Guillarmou, Power spectrum of the geodesic flow on hyperbolic manifolds, Analysis and PDE 8, 923 (2015)
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Lecian, O. (2023). Reduced 2-dimensional Birkhoff surfaces of section of the desymmetrized P SL(2, Z) group: the Anosov characterization. International Journal Of Mathematics And Computer Research, 11(2), 3261-3267. https://doi.org/10.47191/ijmcr/v11i2.07