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Ergodic Theory, Open Dynamics, and Coherent Structures pp 31–39Cite as

Eigenvalues of Transfer Operators for Dynamical Systems with Holes

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 70))

Abstract

For real-analytic expanding open dynamical systems in arbitrary finite dimension, we establish rigorous explicit bounds on the eigenvalues of the corresponding transfer operators acting on spaces of holomorphic functions. In dimension 1 the eigenvalue decay rate is exponentially fast, while in dimension d it is \(O{(\theta }^{{n}^{1/d} })\) as n → ∞ for some 0 < θ < 1.

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Notes

  1. 1.

    When acting on these spaces, \(\mathcal{P}_{H}\) has a strictly positive spectral radius δ, with δ > 0 an eigenvalue such that − logδ is the corresponding escape rate (see, e.g. [14] for one-dimensional maps); thus escape is at an exponential rate, rather than anything faster. Moreover, \({\delta }^{-n}\mathcal{P}_{H}^{n}1 \rightarrow \varrho\), where \(\varrho\) is the density function for the Pianigiani-Yorke measure [15].

  2. 2.

    The study of transfer operators on this space U(D) was inaugurated by Ruelle [18].

  3. 3.

    Ruelle [18], following Grothendieck [11], stated the asymptotics were O(θ n) as n → ∞, independent of the dimension d, though Fried [10] corrected this to \(O{(\theta }^{{n}^{1/d} })\). One novelty of our results, relative to Fried and Ruelle, is that the constant θ, as well as the implicit constant in the big-O asymptotics, is rendered explicit.

  4. 4.

    This is a Banach space version of Weyl’s original inequality [19] in Hilbert space; the constant n n∕2 is optimal (see [12]).

  5. 5.

    As always, we are making the standing assumption that D is an admissible domain, i.e. that the closure of \(\cup _{i\in \mathcal{J}}T_{i}(D)\) lies in D.

  6. 6.

    If D has C 2 boundary, then H 2(D) can be identified with the \({L}^{2}(\partial D,\sigma )\)-closure of U(D), where σ denotes (2d − 1)-dimensional Lebesgue measure on the boundary \(\partial D\), normalised so that σ(∂ D) = 1. The inner product in H 2(D) is given (see [13, Chaps. 1.5 and 8]) by \((f,g) =\int _{\partial D}{f}^{{\ast}}\,\overline{{g}^{{\ast}}}\,d\sigma\), where, for h ∈ H 2(D), the symbol h ∗ denotes the corresponding nontangential limit function in L 2(∂ D, σ).

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Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    Oscar F. Bandtlow & Oliver Jenkinson

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  1. Oscar F. Bandtlow
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  2. Oliver Jenkinson
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Correspondence to Oscar F. Bandtlow .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Leicestershire, Leicestershire, United Kingdom

    Wael Bahsoun

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    Christopher Bose

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Gary Froyland

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Bandtlow, O.F., Jenkinson, O. (2014). Eigenvalues of Transfer Operators for Dynamical Systems with Holes. In: Bahsoun, W., Bose, C., Froyland, G. (eds) Ergodic Theory, Open Dynamics, and Coherent Structures. Springer Proceedings in Mathematics & Statistics, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0419-8_2

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