Dynamical Systems Seminar

Stable limit laws for observables on dynamical systems have been established in two settings: 'good observables' (typically Hölder) on slowly mixing non-uniformly hyperbolic systems and 'bad observables' (unbounded with fat tails) on fast mixing dynamical systems. Consider a polynomially mixing billiard \((T, Q, \mu)\). We use a Poisson point process approach to prove distributional convergence to a stable law for non-square integrable observables \(\varphi(x)= d(x,x_0)^{-\frac{2}{\alpha}}, 0< \alpha < 2,\) and \(x_{0} \in Q\). The observable \(\varphi\) has a tail distribution of index \(\alpha\), i.e. \(\mu(\varphi>t) \sim t^{-\alpha}\). The billiard systems we consider have a sufficiently slow mixing rate that Hölder observables satisfy a stable law of index \(\gamma\), where \(\gamma\) is a mixing parameter. We will investigate the interplay between the stable law arising from the 'fat tailed' observable and the stable law arising from the slow mixing dynamics. A key result is to verify a standard mixing condition, which ensures that large values of the observable dominate the time-series, in the range \(1<\alpha<2\).