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Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions
Graduate School of Advanced Science and Engineering, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan |
Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.
References:
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V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 29–68.
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D. Fried,
The zeta functions of Ruelle and Selberg. I, Ann. Sci. École. Norm. Sup. (4), 19 (1986), 491-517.
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N. T. A. Haydn,
Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynam. Systems, 10 (1990), 347-360.
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M. Jézéquel,
Local and global trace formulae for smooth hyperbolic diffeomorphisms, J. Spectr. Theory, 10 (2020), 185-249.
doi: 10.4171/JST/290. |
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H. König, Eigenvalue Distribution of Compact Operators, Operator Theory: Advances and Applications, 16, Birkhäuser Verlag, Basel, 1986.
doi: 10.1007/978-3-0348-6278-3. |
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W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990). |
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A. Pietsch, Eigenvalues and $s$-Numbers, Cambridge Studies in Advanced Mathematics, 13, Cambridge University Press, Cambridge, 1987.
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M. Pollicott,
Meromorphic extensions of generalised zeta functions, Invent. Math., 85 (1986), 147-164.
doi: 10.1007/BF01388795. |
| [11] |
A. Quas and J. Siefken,
Ergodic optimization of super-continuous functions on shift spaces, Ergodic Theory Dynam. Systems, 32 (2012), 2071-2082.
doi: 10.1017/S0143385711000629. |
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D. Ruelle,
Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.
doi: 10.1007/BF01403069. |
| [13] |
D. Ruelle,
An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math., 72 (1990), 175-193.
doi: 10.1007/BF02699133. |
| [14] |
F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
|
show all references
References:
| [1] |
V. Baladi and M. Tsujii, Dynamical determinants and spectrum for hyperbolic diffeomorphisms, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 29–68.
doi: 10.1090/conm/469/09160. |
| [2] |
R. P. Boas, Entire Functions, Academic Press Inc., New York, 1954. |
| [3] |
M. Demuth, F. Hanauska, M. Hansmann and G. Katriel,
Estimating the number of eigenvalues of linear operators on Banach spaces, J. Funct. Anal., 268 (2015), 1032-1052.
doi: 10.1016/j.jfa.2014.11.007. |
| [4] |
D. Fried,
The zeta functions of Ruelle and Selberg. I, Ann. Sci. École. Norm. Sup. (4), 19 (1986), 491-517.
doi: 10.24033/asens.1515. |
| [5] |
N. T. A. Haydn,
Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynam. Systems, 10 (1990), 347-360.
doi: 10.1017/S0143385700005587. |
| [6] |
M. Jézéquel,
Local and global trace formulae for smooth hyperbolic diffeomorphisms, J. Spectr. Theory, 10 (2020), 185-249.
doi: 10.4171/JST/290. |
| [7] |
H. König, Eigenvalue Distribution of Compact Operators, Operator Theory: Advances and Applications, 16, Birkhäuser Verlag, Basel, 1986.
doi: 10.1007/978-3-0348-6278-3. |
| [8] |
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990). |
| [9] |
A. Pietsch, Eigenvalues and $s$-Numbers, Cambridge Studies in Advanced Mathematics, 13, Cambridge University Press, Cambridge, 1987.
|
| [10] |
M. Pollicott,
Meromorphic extensions of generalised zeta functions, Invent. Math., 85 (1986), 147-164.
doi: 10.1007/BF01388795. |
| [11] |
A. Quas and J. Siefken,
Ergodic optimization of super-continuous functions on shift spaces, Ergodic Theory Dynam. Systems, 32 (2012), 2071-2082.
doi: 10.1017/S0143385711000629. |
| [12] |
D. Ruelle,
Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.
doi: 10.1007/BF01403069. |
| [13] |
D. Ruelle,
An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math., 72 (1990), 175-193.
doi: 10.1007/BF02699133. |
| [14] |
F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
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