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Smooth expanding maps: Dynamical determinants

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Book cover Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

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Abstract

The main result of this chapter is a variant of Ruelle’s theorem on the dynamical determinants of transfer operators associated with differentiable expanding dynamics and weights. The proof uses the Milnor-Thurston kneading operator approach. The contents of this chapter are a blueprint for the technically more involved situation of hyperbolic dynamics and the corresponding anisotropic Banach spaces in Part II.

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Notes

  1. 1.

    A power of a formal power series is a formal power series, the exponential of a formal power series thus gives a formal power series, using the Taylor series at zero of the exponential function.

  2. 2.

    The case \(d=1\) was considered in the introduction.

  3. 3.

    In particular, the domain of analyticity obtained in Theorem 3.3 is not optimal for general nonlinear expanding dynamics.

  4. 4.

    If \(C>1\), different arguments will be used in Section 3.3 if \(\alpha>d+t\) and in Section 3.3.4 in the more difficult case of low differentiability \(\alpha\le d+t\).

  5. 5.

    If \(L>1\), different arguments will be used in Sections 3.3 and 3.3.4.

  6. 6.

    See the remark after Proposition 3.13.

  7. 7.

    A rate of convergence is given by \(\| \mathbf{I}_{\epsilon}(\varphi)-\varphi\|_{H^{t}_{p}(M)} \le C \epsilon^{t-t'} \|\varphi\|_{H^{t'}_{p}(M)} \) for all \(0\le t'< t\), see e.g. [25, Lemma 5.4].

  8. 8.

    The decomposition is independent of \(t\) and \(p\).

  9. 9.

    At the end of this section, we give an alternative proof, using regularised determinants, in which the kneading operator is explicited.

  10. 10.

    Problem 2.46 would allow us to simplify the argument somewhat. Note also that a finite matrix of operators, indexed by \(\omega\), as in [100] can further streamline the proof without requiring a countable matrix as in [31]. These remarks also apply, for instance, to the proof of Proposition 3.18, and to hyperbolic settings.

  11. 11.

    As in the proof of Proposition 3.18, we can safely ignore the operators \(A_{t}\) there.

  12. 12.

    If \(\omega_{j}\ne \omega'(\overrightarrow{\omega}_{j})\), we may proceed as in the proof of Proposition 3.18, see (3.61).

  13. 13.

    As this book was going to press, M. Jézéquel [101] announced a series of new examples of non-polar singularities.

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  1. IMJ-PRG, Sorbonne Université and CNRS, Paris, France

    Viviane Baladi

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Baladi, V. (2018). Smooth expanding maps: Dynamical determinants. In: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 68. Springer, Cham. https://doi.org/10.1007/978-3-319-77661-3_3

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