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.
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.
The case \(d=1\) was considered in the introduction.
- 3.
In particular, the domain of analyticity obtained in Theorem 3.3 is not optimal for general nonlinear expanding dynamics.
- 4.
- 5.
- 6.
See the remark after Proposition 3.13.
- 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.
The decomposition is independent of \(t\) and \(p\).
- 9.
At the end of this section, we give an alternative proof, using regularised determinants, in which the kneading operator is explicited.
- 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.
As in the proof of Proposition 3.18, we can safely ignore the operators \(A_{t}\) there.
- 12.
- 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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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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