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Smooth expanding maps: The spectrum of the transfer operator

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Abstract

This chapter presents a variant of Ruelle’s bound on the essential spectral radius of transfer operators associated with differentiable expanding dynamics and weights, replacing the Hölder spaces by Sobolev spaces. The chapter ends with the Gouëzel-Keller-Liverani perturbation theory, which will also be applicable in the hyperbolic setting of Part II.

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Notes

  1. 1.

    We shall not discuss the analytic case, referring instead to [137], [73, 150, 151], [74, 1], and [160, 161].

  2. 2.

    Existence of a maximal eigenvalue can also be obtained from Theorems 3.3 or 3.5 and Corollary 3.8, as explained in Lemma 6.1 in the hyperbolic case.

  3. 3.

    See Collet–Isola [52] for an earlier result in the symbolic setting for \(0< t<1\).

  4. 4.

    See [52, 107] for eigenvalues of modulus \(\le\ell^{-r}\), in slightly different settings.

  5. 5.

    Section 7.1.1 of the present book contains the analogous theory for the peripheral spectrum of hyperbolic maps \(T\).

  6. 6.

    This follows from the fact that expanding maps are topologically mixing.

  7. 7.

    See Appendix B. We use the expressions equilibrium state and equilibrium measure interchangeably.

  8. 8.

    \(\Pi _{j}\) is just the spectral projector \((2\pi \mathrm{i})^{-1}\int_{|z-\gamma_{j}|=\epsilon_{j}} (z-\mathcal{L}_{g})^{-1} \, \mathrm{d}z\) for the isolated eigenvalue \(\gamma_{j}\).

  9. 9.

    Spaces of bounded variation or generalised bounded variation can also be used, but this approach is not simpler to carry out.

  10. 10.

    See e.g. [165, §I.6, Chap. XI], where the notation \(\mathcal{L}^{t}_{p}\) is used.

  11. 11.

    Since \(\Omega\) is finite, the norms defined by taking the sum or the maximum over \(\omega\) are equivalent. In order to get a Hilbert norm when \(p=2\), one should consider the sum.

  12. 12.

    See also [40].

  13. 13.

    See the discussion before Theorem 2.31 below and Appendix D.1 for the general notion of an operator \(a^{Op}\) associated with a symbol \(a\).

  14. 14.

    The limit (2.27) exists and coincides with the infimum by submultiplicativity.

  15. 15.

    Note that the last claim of [11, Lemma 16 (3)] is incorrect in general, the number \(\theta\) there must be replaced by the maximal contraction.

  16. 16.

    The extendability condition (2.29) holds e.g. if \(U\) is a small ball and \(F\) is close enough to its derivative on \(U\). Condition (2.29) is not needed in Section 2.3.2. It will be used to control the term \(\mathcal{M}_{c}\) in the decomposition (2.95) in Section 2.4.2.

  17. 17.

    The indicator function of a set \(X\) satisfies \(\mathbf{1}_{X}^{p/(p-1)}=\mathbf{1}_{X}\) for all \(1< p<\infty\).

  18. 18.

    A topological space in which distinct points have disjoint neighbourhoods.

  19. 19.

    This can be done by using a cover of \(M\) by \(d\)-dimensional balls of radius equal to the Lebesgue number of \(\mathcal{V}_{m}\) and with centers on an appropriate lattice; see e.g. [179] for a more general result, and the references therein. It is possible to take \(\nu_{d}=d+1\), see [176, Thm 4.5.8, Thm 4.5.13].

  20. 20.

    Together with Remark 2.22, up to replacing \(\tilde{g}\) by \(\tilde{g}+\epsilon\) for arbitrarily small \(\epsilon\) if \(t'\) is very small

  21. 21.

    A smooth expanding map has no complexity at the beginning, in contrast to the piecewise smooth systems in [167] or [47].

  22. 22.

    In particular, \(\psi_{n}\) is a symbol of order 0 (see Appendix D.1).

  23. 23.

    See the discussion before Theorem 2.31 below and Appendix D.1 for the general notion of an operator \(a^{Op}\) associated with a symbol \(a\).

  24. 24.

    This is basically a generalisation of Pythagoras’ theorem.

  25. 25.

    An analogous decomposition will be essential in Chapter 3 and in the hyperbolic setting of Part II.

  26. 26.

    In the hyperbolic setting of Part II, the analogue of Lemma 2.34 is (4.57).

  27. 27.

    The finite set depends on the dimension \(d\).

  28. 28.

    The argument above also shows that there exists a constant \(C>0\) such that \(b_{n} * b_{m}(x)\le C \cdot b_{\min\{n,m\}}(x)\) for any \(x\in \mathbb{R}^{d}\) and any \(n,m\ge0\), as noted in [31, (4.21)]. The proof can be organised differently, stating this fact as a separate lemma, see [31, App C].

  29. 29.

    If \(r\) is large enough and \(T_{\epsilon}\in C^{N}([-1,1],C^{r}(M,M))\) for \(N\ge3\), it is possible to write down explicit formulas for derivatives of higher order.

  30. 30.

    The invariant density has singularities of the type \(\sqrt{x-c_{k}}^{-1}\), and belongs to \(H^{t}_{p}\) for \(t<1/2\) and \(p>1\) close enough to 1.

  31. 31.

    By (2.25) in Lemma 2.16, the upper bound \(R_{*}^{t,p}(g)\) for \(p\ne\infty\) improves the bound in [11]. See also footnote 15.

  32. 32.

    There, a dynamical proof is given, bypassing Theorem C.1.

  33. 33.

    See also [84, Thm 3.3] for a slight generalisation.

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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: The spectrum of the transfer operator. 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_2

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