Duke Mathematical Journal

Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions

Mathieu Baillif

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We study transfer operators $\mathcal{M}^{(k)}$ associated to a finite family {ψω} of $\mathcal{C}^{r}$ (r ≥ 1) transversal maps Uω$\mathbb{R}^n, where Uω$\mathbb{R}^n, with $\mathcal{C}^{r}$ compactly supported weights gω, acting on k-forms in $\mathbb{R}^n. Using the definitions of sharp trace Tr and flat trace Tr, the following formula holds between power series: $\Det^{\#}(1-z\mathcal{M})=\Pi_{k=0}^n \Det^{\flat}(1-z\mathcal{M}^{(k)})^{(-1)^k}$. Following ideas of Kitaev [17], we define kneading operators $\mathcal{D}_k$(z), which are kernel operators. Our main result is the equality (as formal power series)

$$ \Det^{\#}(1-z\mathcal{M})=\prod_{k=0}^{n-1} \Det^{\flat}\big(1+\mathcal{D}_k(z)\big)^{(-1)^{k+1}}.$$

We also show that a finite power of $\mathcal{D}_k$(z) is trace-class on L2. This (partially) generalizes results obtained by Baladi, Kitaev, Ruelle, and Semmes in dimension one, complex and real [8], [10]).

Article information

Duke Math. J. Volume 124, Number 1 (2004), 145-175.

First available in Project Euclid: 30 July 2004

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Zentralblatt MATH identifier

Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems


Baillif, Mathieu. Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions. Duke Math. J. 124 (2004), no. 1, 145--175. doi:10.1215/S0012-7094-04-12415-7. https://projecteuclid.org/euclid.dmj/1091217477

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  • M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes, I, Ann. of Math. (2) 86 (1967), 374--407.
  • --. --. --. --., A Lefschetz fixed point formula for elliptic complexes, II: Applications, Ann. of Math. (2) 88 (1968), 451--491.
  • --. --. --. --., ``Notes on the Lefschetz fixed point formula for elliptic complexes'' in Raoul Bott: Collected Papers, Vol. 2: Differential Operators, Contemp. Mathematicians, Birkhäuser, Boston, 1994, 68--162.
  • M. Baillif and V. Baladi, Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case, preprint.
  • V. Baladi, Infinite kneading matrices and weighted zeta functions of interval maps, J. Funct. Anal. 128 (1995), 226--244.
  • --. --. --. --., Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems 18 (1998), 255--292.
  • --------, Positive Transfer Operators and Decay of Correlations, Adv. Ser. Nonlinear Dynam. 16, World Sci., River Edge, N.J., 2000.
  • V. Baladi, A. Kitaev, D. Ruelle, and S. Semmes, Sharp determinants and kneading operators for holomorphic maps, Proc. Steklov Inst. Math. 216, no. 1 (1997), 186--228.
  • V. Baladi and D. Ruelle, An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps, Ergodic Theory Dynam. Systems 14 (1994), 621--632.
  • --. --. --. --., Sharp determinants, Invent. Math. 123 (1996), 553--574.
  • I. Gohberg, S. Goldberg, and N. Krupnik, Traces and Determinants of Linear Operators, Oper. Theory Adv. Appl. 116, Birkhäuser, Basel, 2000.
  • S. Gouëzel, Spectre de l'opérateur de transfert en dimension $1$, Manuscripta Math. 106 (2001), 365--403.
  • V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977.
  • G. Henkin and J. Leiterer, Theory of Functions on Complex Manifolds, Monogr. Math. 79, Birkhäuser, Basel, 1984.
  • L. Hörmander, ``Spectral analysis of singularities'' in Seminar on Singularities of Solutions of Linear Partial Differential Equations (Princeton, 1977/78), ed. L. Hörmander, Ann. Math. Stud. 91, Princeton Univ. Press, Princeton, 1979.
  • V. Yu. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys. 211 (2000), 253--271.
  • A. Kitaev, Kneading operators in higher dimensions, private communication, June 1995.
  • J. Milnor and W. Thurston, ``On iterated maps of the interval'' in Dynamical Systems (College Park, Md., 1986--87.), ed. J. C. Alexander, Lecture Notes in Math. 1342, Springer, Berlin, 1988.
  • D. Ruelle, ``Sharp zeta functions for smooth interval maps'' in International Conference on Dynamical Systems: A Tribute to Ricardo Mañé (Montevideo, Uruguay, 1995), ed. F. Ledrappier, J. Lewowicz, and S. Newhouse, Pitman Res. Notes Math. Ser. 362, Longman, Harlow, England, 1996, 188--206.
  • --------, Functional determinants related to dynamical systems and the thermodynamic formalism, Fermi lectures, Scuola Norm. Sup., Pisa, preprint, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1995.
  • L. Schwartz, Théorie des distributions, new ed., Publ. Inst. Math. Univ. Strasbourg 9--10, Hermann, Paris, 1966.
  • B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge Univ. Press, Cambridge, 1979.
  • M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Adv. Ser. Nonlinear Dynam., Benjamin, New York, 1965.