We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps using the full parameter range , in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e. smallest α ) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (theorem 1.1) establishes sharp estimates on the position of return time intervals for the quenched dynamics. The main applications of this estimate are to limit laws (in particular, CLT and stable laws, depending on the parameters chosen in the range ) for the associated skew product; these are detailed in theorem 3.2. Since our estimates in theorem 1.1 also hold for we study a second class of random transformations derived from piecewise affine Gaspard–Wang maps, prove existence of an infinite ( σ -finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in <!-- MATH $[\alpha _0,\alpha _1]\subset (0,1)$ --> chosen independently with respect to a distribution on <!-- MATH $[\alpha _0,\alpha _1]$ --> and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every 0$">, for almost every <!-- MATH $\omega \in [\alpha _0,\alpha _1]^\mathbb{Z}$ --> , the upper bound <!-- MATH $n^{1-\frac {1}{\alpha _0}+\delta }$ --> holds on the rate of decay of correlation for Hölder observables on the fibre over . For three different distributions on <!-- MATH $[\alpha _0,\alpha _1]$ --> (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from <!-- MATH $n^{-\frac {1}{\alpha _0}}$ --> to <!-- MATH $(\log n)^{\frac {1}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$ --> to <!-- MATH $(\log n)^{\frac {2}{\alpha _0}}\cdot n^{-\frac {1}{\alpha _0}}$ --> , respectively.

We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate ${C}^{\ast } \cdot (\ln m)/ m$ , where ${C}^{\ast } $ is a computable fixed constant and ${m}^{- 1} $ is the mesh size of the discretization.

We correct a mistake in the definition of the skew product (4.1) of [1].

We establish some statistical properties of the hyperbolic times for a class of non-uniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric baker’s map-type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last 30 years. The statistics of these maps are controlled by the order of tangency (linked to a single parameter α, where 0 < α < ∞) that a certain ‘cut function’ makes with the boundary of the square. Previously, a direct Young tower construction has been used to obtain optimal correlation decay rates of O(n^−1/α) for Hölder observables and all values of the parameter α. A central limit theorem (CLT) is obtained when 0 < α < 1. The asymptotics of a natural hyperbolic time for this family of maps are analysed via the same Young tower. By using a large deviations result of Melbourne and Nicol, we prove that the first hyperbolic time is integrable if and only if the parameter satisfies 0 < α < 1. Furthermore, within this restricted range of parameters, concentration inequalities recently established by Chazottes and Gouëzel imply sharp O(n^−1/α) bounds on the tail distribution of first hyperbolic times. As shown by Alves, Viana, and others, knowledge of the tail distribution of the hyperbolic times leads to upper bounds on the rate of decay of correlations and derivation of a CLT. Comparing to the results obtained directly for this family of maps, the latter estimates via hyperbolic times are suboptimal, even over the restricted range of parameters 0 < α < 1.

In this article we study a piecewise linear discretization schemes for transfer operators (Perron-Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous {\bf pointwise} approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the $L^{\infty}$-norm. The outcome of this paper complements rigorous results on $L^1$ approximations of invariant densities \cite{KMY} and recent results on the formulae of escape rates of open dynamical systems \cite{KL2}. We implement our computations on two examples (one rigorous and one non-rigorous) to illustrate the feasibility and efficiency of our schemes.