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Given , the interval map given by is naturally semiconjugate to the shift . The shift space admits a natural description in terms of the lexicographic order, and another in terms of a countablestate directed graph. These have been used … Continue reading
1. Sgap shifts Sgap shifts are a useful example for studying dynamics of shift spaces that are not subshifts of finite type but still exhibit some strong mixing properties. They are defined as follows: given , let be the set … Continue reading
There are two equivalent definitions of mixing for a measurepreserving dynamical system . One is in terms of sets: for all measurable . The other is in terms of functions: for all . In both cases one may refer to … Continue reading
Two very different types of dynamical behaviour are illustrated by a pair of very wellknown examples on the circle: the doubling map and an irrational rotation. On the unit circle in , the doubling map is given by , while … Continue reading
An important issue in hyperbolic dynamics is that of absolute continuity. Suppose some neighbourhood of a smooth manifold is foliated by a collection of smooth submanifolds , where is some indexing set. (Here “smooth” may mean , or , or … Continue reading
This week’s post continues last week’s discussion of Markov chains and mixing times, and introduces the idea of coupling as a method for estimating mixing times. We remark that some nice notes on the subject of coupling (and others) can … Continue reading
Our seminar series is taking a hiatus from spectral methods for a couple weeks — these will return eventually, but in the meantime we’ll spend some time with the idea of coupling as a method for deriving statistical properties. In … Continue reading