Parabolic dynamical systems and inducing. (Q2751497)

In this paper, the author tries to sketch a somewhat general procedure to approach spectral properties of dynamical systems in situations where the usual hyperbolicity assumptions are partially relaxed, such as parabolic maps of the Riemann sphere.NEWLINENEWLINENEWLINENEWLINEMore precisely, let \((X,T,m)\) be a triple where \(T\) is a noninvertible transformation of \(X\) and \(m\) is a finite reference measure. Assume also that \(T\) is locally invertible. The usual tool to study spectral properties of \(T\) is the transfer operator \(P\) and more generally \(P_t\). Let \(\Delta\) be a subset of \(X\), and assume \(T\) is nonexpanding outside \(\Delta\). Let \(R\) be the first passage function in \(\Delta\) and denote the induced transformation by \(T_R\). Consider the operator \(\mathcal L_{t,z}\) formally defined by NEWLINE\[NEWLINE\mathcal L_{t,z}f(x)=\sum_{y\colon T_R(y)=x}z^{R(y)}\frac{f(y)}{|DT_R(y)|^t}.NEWLINE\]NEWLINE In this paper, the author gives some conditions for the operator \(\mathcal L_{t,z}\) to be well defined and then he relates its spectral properties to those of \(P_t\). In the second part, he explains the results he obtains using these spectral properties in the case of parabolic rational maps: analytic properties of the pressure, escape rate and asymptotic distribution of the preimages.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00028].