Decay of correlations for piecewise smooth maps with indifferent fixed points.

*(English)*Zbl 1071.37026Let \(f : [0, 1] \to [0, 1]\) be a piecewise smooth map which is expanding on \([0, 1]\) and has an indifferent fixed-point at \(0\). It is assumed that near \(0\), \(f\) has the form \(f (x) = x + x^{1 + \gamma} + x^{1 + \gamma} \sigma_0 (x)\) with \(\sigma_0 (x) \to 0\) as \(x \to 0\) together with some additional constraint on the derivative of \(f\) at \(0\). The author first shows that there is a finite absolutely continuous invariant measure \(\mu\) for \(f\) of the form \(d_\mu = h \, dx\) with \(\lim_{x \to 0} x^\gamma h (x) = G_0\) for a constant \(G_0 \geq 0\). This fact was derived by G. Pianigiani [Isr. J. Math. 35, 32–48 (1980; Zbl 0445.28016)]; the proof given by the author uses the Perron-Frobenius operator \(\mathcal L : C^0 [0, 1] \to C^0 [0, 1]\) defined by \(\mathcal L g (x) = \sum_{y \in f^{-1} (x)} \frac{g(y)}{f^\prime (y)}\) together with some lengthy estimates (to show that \(\mathcal L\) is continuous for a suitable norm on \(C^0 [0, 1]\)).

The main result of the paper shows that the operator \(f\) has polynominal decay of correlations with rate of order \(n^{1 - \beta}\) where \(\beta = \gamma^{-1}\) for Lipschitz observables. This estimate is shown to be sharp in the sense that there are Lipschitz functions \(F\) and \(G\) on \([0, 1]\) and a constant \(c^\prime > 0\) such that \(| \int (F \circ f^n)\, G\, d_\mu - \int F d_\mu \cdot \int G d_\mu | \geq \frac{C^\prime}{n^{\beta-1}}\) for all \(n \geq 0\). As before, the proof consists in a careful analysis of a suitably chosen Perron-Frobenius operator.

The main result of the paper shows that the operator \(f\) has polynominal decay of correlations with rate of order \(n^{1 - \beta}\) where \(\beta = \gamma^{-1}\) for Lipschitz observables. This estimate is shown to be sharp in the sense that there are Lipschitz functions \(F\) and \(G\) on \([0, 1]\) and a constant \(c^\prime > 0\) such that \(| \int (F \circ f^n)\, G\, d_\mu - \int F d_\mu \cdot \int G d_\mu | \geq \frac{C^\prime}{n^{\beta-1}}\) for all \(n \geq 0\). As before, the proof consists in a careful analysis of a suitably chosen Perron-Frobenius operator.

Reviewer: Ursula Hamenstädt (Bonn)

##### MSC:

37E05 | Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) |

37A25 | Ergodicity, mixing, rates of mixing |