Abstract
We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.
Highlights
The setting of this paper is of skew product systems of interval maps over linearly expanding interval mapsG (y, x) = (Em(y), gy(x)) on I × [0, 1]
Is there a setting in which to understand both the random and the chaotic driving cases? We suggest that one approach to answering this question is through the study of ‘chaotic walks’, i.e. additive walks where the increments are chosen from some chaotic process. [· · ·] We know of no systematic studies of chaotic walks, though they are clearly an important counterpart to the comparatively well-studied random walks
In [11] the reader can find a study of iterated function systems of interval diffeomorphisms with a neutral fixed point
Summary
In [11] the reader can find a study of iterated function systems of interval diffeomorphisms with a neutral fixed point This amounts to a study of chaotic walks on the line: for instance the symmetric random walk can be cast as a skew product setting of the form (y, x). A similar result can be formulated for a skew product G ∈ Swith vanishing fiber Lyapunov exponents at both boundaries, i.e. where L1 = 0 in (4): just replace U by an interval [ , 1 − ] with > 0 small. We will approximate G ∈ S with step skew product systems driven by subshifts of finite type This is done by using fine Markov partitions for E2 of I. It induces a subshift of finite type on sequences with four symbols 1, 2, 3, 4
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