One-dimensional maps via complex analysis in several variables. (English) Zbl 0879.58023
Let \(f:I\to I\) be a mapping of an interval \(I\) that is piecewise continuous and monotone with respect to a finite partition \({\mathcal I}\) of \(I\). Let \(g:I \to \mathbb{R}^+\) be a bounded function which is continuous on each element of \({\mathcal I}\) and such that the variation \(\text{var} (g^\alpha) <\infty\) for each \(\alpha>0\). Let \(z,s \in\mathbb{C}\).
The author studies the zeta function \[ \zeta (z,s)= \exp \left(\sum^\infty_{n=1} {z^n \over n} \sum_{x=f^n(x)} \bigl(g_n(x) \bigr)^s \right) \] where \(g_n(x)= \prod^{n-1}_{i=0} g(f^ix)\), and transfer operators \[ L_{g^s} k(x)= \sum_{y\in f^{-1}x} \bigl(g(y) \bigr)^s k(y) \] acting on functions \(k:I \to\mathbb{C}\). Particular attention is paid to the case \(g(x)= |Df(x) |\), to which we restrict in this review. For fixed \(s\) with \({\mathfrak R} s>0\) the correspondence between poles \(z\) of the zeta function and eigenvalues \(z^{-1}\) of the transfer operator was proved in [V. Baladi and G. Keller, Commun. Math. Phys. 127, No. 3, 459-477 (1990; Zbl 0703.58048)]. Using complex continuation methods in several variables, D. Ruelle [Helv. Phys. 66, 181-191 (1993; Zbl 0829.58033)] extended the meromorphy properties of the zeta function to \(s\in \mathbb{C}\) with \({\mathfrak R} s<0\).
In this paper analogous arguments are applied to the resolvents \(R(z,s)\) of \(L_{g^s}\) interpreted as operators from the space of functions of bounded variation to \(L^1_{\text{Leb}}\). This allows, among other things, the following conclusion: Suppose that \(f\) is uniformly hyperbolic on periodic points and that \(\lim \sup_{n\to \infty} (\int{1 \over |Df^n |^\varepsilon} dx)^{1/n} <1\) for some \(\varepsilon >0\). Then each weakly mixing absolutely continuous \(f\)-invariant measure has exponentially decreasing correlations. Similarly, detailed results about the distribution of periodic orbits are derived, and in an appendix the case of \(C^r\) Markov maps on \(I\) is discussed, where better estimates are possible due to the stronger smoothness assumption on the map. (For related results see also [G. Keller and T. Nowicki, Commun. Math. Phys. 149, No. 1, 31-69 (1992; Zbl 0763.58024)]).
The author studies the zeta function \[ \zeta (z,s)= \exp \left(\sum^\infty_{n=1} {z^n \over n} \sum_{x=f^n(x)} \bigl(g_n(x) \bigr)^s \right) \] where \(g_n(x)= \prod^{n-1}_{i=0} g(f^ix)\), and transfer operators \[ L_{g^s} k(x)= \sum_{y\in f^{-1}x} \bigl(g(y) \bigr)^s k(y) \] acting on functions \(k:I \to\mathbb{C}\). Particular attention is paid to the case \(g(x)= |Df(x) |\), to which we restrict in this review. For fixed \(s\) with \({\mathfrak R} s>0\) the correspondence between poles \(z\) of the zeta function and eigenvalues \(z^{-1}\) of the transfer operator was proved in [V. Baladi and G. Keller, Commun. Math. Phys. 127, No. 3, 459-477 (1990; Zbl 0703.58048)]. Using complex continuation methods in several variables, D. Ruelle [Helv. Phys. 66, 181-191 (1993; Zbl 0829.58033)] extended the meromorphy properties of the zeta function to \(s\in \mathbb{C}\) with \({\mathfrak R} s<0\).
In this paper analogous arguments are applied to the resolvents \(R(z,s)\) of \(L_{g^s}\) interpreted as operators from the space of functions of bounded variation to \(L^1_{\text{Leb}}\). This allows, among other things, the following conclusion: Suppose that \(f\) is uniformly hyperbolic on periodic points and that \(\lim \sup_{n\to \infty} (\int{1 \over |Df^n |^\varepsilon} dx)^{1/n} <1\) for some \(\varepsilon >0\). Then each weakly mixing absolutely continuous \(f\)-invariant measure has exponentially decreasing correlations. Similarly, detailed results about the distribution of periodic orbits are derived, and in an appendix the case of \(C^r\) Markov maps on \(I\) is discussed, where better estimates are possible due to the stronger smoothness assumption on the map. (For related results see also [G. Keller and T. Nowicki, Commun. Math. Phys. 149, No. 1, 31-69 (1992; Zbl 0763.58024)]).
Reviewer: G.Keller (Erlangen)
MSC:
| 37E99 | Low-dimensional dynamical systems |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 32D05 | Domains of holomorphy |
Keywords:
interval maps; logistic map; Bochner theorem; zeta function; Ruelle-Perron-Frobenius operator; transfer operatorReferences:
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