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

Katsukuni Nakagawa

doi:10.3934/dcds.2020282

Article Contents

2020, Volume 40, Issue 11: 6331-6350. Doi: 10.3934/dcds.2020282

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

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: November 30, 2019

Revised: April 30, 2020

Published: July 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37C30, 37B10; Secondary: 47B06.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.