Thermodynamic Formalism and Quantum Mechanics on the Modular Surface

Mayer, Daniel

doi:10.1142/9789814355872_0039

Cited by 4 publications

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“…where x φ is an arbitrary point of φ. On the other hand according to Series [2], Adler and Flatto [1] there is an one to one correspondence between P er(r) and the set of primitive periodic orbits ϑ on the unit tangent bundle T 1 M , M = P SL(2, Z)\H, with the period (see [12])…”

Section: Selberg Zeta Function and Transfer Operatormentioning

confidence: 99%

Mayer Transfer Operator Approach to Selberg Zeta Function

Momeni,

Venkov

2010

Preprint

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Denmark (December 2009). We mostly present here a survey of results of Dieter Mayer on relations between Selberg and Smale-Ruelle dynamical zeta functions. In a special situation the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions and its matrix representation in a natural basis is given in terms of the Riemann zeta function and the Euler gamma function.

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Section: Selberg Zeta Function and Transfer Operatormentioning

confidence: 99%

Mayer Transfer Operator Approach to Selberg Zeta Function

Momeni,

Venkov

2010

Preprint

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“…(Dynamical Fredholm determinants and transfer operators are also useful in quantum chaos, at the triple intersection of number theory, geometry, and group theory. Mayer [73] wrote very readable discussion on the Selberg zeta function and transfer operators applied to quantum chaos on the modular surface. See also [32] and references therein.…”

Section: Dynamical Zeta Functions and Dynamical Fredholm Determinantsmentioning

confidence: 99%

Spectrum and Statistical Properties of Chaotic Dynamics

Baladi

2001

European Congress of Mathematics

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Abstract. We present new developments on the statistical properties of chaotic dynamical systems. We concentrate on the existence of an ergodic physical (SRB) invariant measure and its mixing properties, in particular decay of its correlation functions for smooth observables. In many cases, there is a connection (via the spectrum of a Ruelle-Perron-Frobenius transfer operator) with the analytic properties of a weighted dynamical zeta function, weighted dynamical Lefschetz function, or dynamical Ruelle-Fredholm determinant, built using the periodic orbit structure of the map.

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“…In Mayer's theory a transfer operator L s is introduced as a special case of Ruelle's operator for a dynamical system, for instance for the group SL(2, Z) this is the twice iterated Gauss map T = T 2 G acting on the unit interval and the weight function −βlog|T ′ (z)|. Then, as a result of the one-to-one correspondence between the closed geodesics on the modular surface M = H \ SL(2, Z) and the primitive periodic orbits of T the Selberg zeta function can be written as a Fredholm determinant of the transfer operator [3].…”

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confidence: 99%

An application of Jacquet-Langlands correspondence to transfer operators for geodesic flows on Riemann surfaces

Momeni,

Venkov

2008

Preprint

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Thermodynamic Formalism and Quantum Mechanics on the Modular Surface

Cited by 4 publications

References 0 publications

Mayer Transfer Operator Approach to Selberg Zeta Function

Mayer Transfer Operator Approach to Selberg Zeta Function

Spectrum and Statistical Properties of Chaotic Dynamics

An application of Jacquet-Langlands correspondence to transfer operators for geodesic flows on Riemann surfaces

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