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November  2020, 40(11): 6331-6350. doi: 10.3934/dcds.2020282

Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions

Katsukuni Nakagawa

Graduate School of Advanced Science and Engineering, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan

Received  December 2019 Revised  May 2020 Published  July 2020

Fund Project: The author is supported by FY2019 Hiroshima University Grant-in-Aid for Exploratory Research (The researcher support of young Scientists)

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.

Keywords: Ruelle zeta functions, transfer operators, trace formula, spectral representation, super-continuous functions.
Mathematics Subject Classification: Primary: 37C30, 37B10; Secondary: 47B06.
Citation: Katsukuni Nakagawa. Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6331-6350. doi: 10.3934/dcds.2020282
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990).  Google Scholar

[9] A. Pietsch, Eigenvalues and $s$-Numbers, Cambridge Studies in Advanced Mathematics, 13, Cambridge University Press, Cambridge, 1987.   Google Scholar
[10]

M. Pollicott, Meromorphic extensions of generalised zeta functions, Invent. Math., 85 (1986), 147-164.  doi: 10.1007/BF01388795.  Google Scholar

[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.  Google Scholar

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D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.  doi: 10.1007/BF01403069.  Google Scholar

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D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math., 72 (1990), 175-193.  doi: 10.1007/BF02699133.  Google Scholar

[14] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.   Google Scholar

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.  Google Scholar

[2]

R. P. Boas, Entire Functions, Academic Press Inc., New York, 1954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Jézéquel, Local and global trace formulae for smooth hyperbolic diffeomorphisms, J. Spectr. Theory, 10 (2020), 185-249.  doi: 10.4171/JST/290.  Google Scholar

[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.  Google Scholar

[8]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990).  Google Scholar

[9] A. Pietsch, Eigenvalues and $s$-Numbers, Cambridge Studies in Advanced Mathematics, 13, Cambridge University Press, Cambridge, 1987.   Google Scholar
[10]

M. Pollicott, Meromorphic extensions of generalised zeta functions, Invent. Math., 85 (1986), 147-164.  doi: 10.1007/BF01388795.  Google Scholar

[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.  Google Scholar

[12]

D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231-242.  doi: 10.1007/BF01403069.  Google Scholar

[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.  Google Scholar

[14] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.   Google Scholar
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