# American Institute of Mathematical Sciences

• Previous Article
Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations
• DCDS Home
• This Issue
• Next Article
Matching for a family of infinite measure continued fraction transformations
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

 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.

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 [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

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
 [1] Olaf Klein. On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2591-2614. doi: 10.3934/dcds.2015.35.2591 [2] Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521 [3] Roland Martin. On simple Igusa local zeta functions. Electronic Research Announcements, 1995, 1: 108-111. [4] Vesselin Petkov, Luchezar Stoyanov. Ruelle transfer operators with two complex parameters and applications. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6413-6451. doi: 10.3934/dcds.2016077 [5] João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465 [6] David Karpuk, Anne-Maria Ernvall-Hytönen, Camilla Hollanti, Emanuele Viterbo. Probability estimates for fading and wiretap channels from ideal class zeta functions. Advances in Mathematics of Communications, 2015, 9 (4) : 391-413. doi: 10.3934/amc.2015.9.391 [7] Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917 [8] Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277 [9] Gregory Beylkin, Lucas Monzón. Efficient representation and accurate evaluation of oscillatory integrals and functions. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4077-4100. doi: 10.3934/dcds.2016.36.4077 [10] Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383 [11] Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629 [12] Hirobumi Mizuno, Iwao Sato. L-functions and the Selberg trace formulas for semiregular bipartite graphs. Conference Publications, 2003, 2003 (Special) : 638-646. doi: 10.3934/proc.2003.2003.638 [13] Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205 [14] Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843 [15] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [16] Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106 [17] Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48 [18] Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure & Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5 [19] Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1721-1735. doi: 10.3934/cpaa.2021038 [20] Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046

2020 Impact Factor: 1.392