Narrow Results

We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. It has been recently shown in [Knight & Munday, 2016] how to apply the standard analytical methods to a piecewise-linear version of the Farey map with holes depending on the associated partition, but their results cannot be obtained in the general case we consider here. To overcome these difficulties we propose here to study approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for approximated holes with vanishing measure. The results suggest that the scaling of the escape rate depends on the “shape” of the approximation, and we show that this is a typical feature of systems with an indifferent fixed point, not an artifact of the particular family we consider.

Considering the thermodynamical formalism of dynamical systems, P. Walters showed that for β-transformations all Lipschitz weights define quasi-compact transfer operators and therefore unique equilibrium measures which additionally have positive entropy and good mixing properties. In this note we generalize this to continuous piecewise monotonic maps of the interval. The case of piecewise monotonic maps with discontinuities remains open.

We consider iterated function schemes that contract on average with place-dependent probabilities. We are interested in generalisations of the central limit theorem, particularly to observations with infinite variance. By studying the spectral properties of an associated one-parameter family of transfer operators acting on an appropriate function space, we prove both a distributional and local limit law with convergence to a stable distribution.

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure.

Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

For an integer $m\ge 2$, let ${{𝒫}}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let ${{ℱ}}_m$ be the class of piecewise linear maps on $I$ with constant slope ${\pm }m$ on each element of ${{𝒫}}_m$. We investigate the effect on mixing properties when $f\in {{ℱ}}_m$ is composed with the interval exchange map given by a permutation $\sigma \in S_N$ interchanging the $N$ subintervals of ${{𝒫}}_N$. This extends the work in a previous paper [N. P. Byott, M. Holland and Y. Zhang, DCDS 33 (2013) 3365–3390], where we considered only the “stretch-and-fold” map $f_{\mathrm{\text{sf}}}(x)=mx\mathrm{\text{mod}}1$.