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We introduce an invariant of cohomology in Bernoulli shifts, which is used to answer a question about cohomology of Hölder functions with finitary functions whose coding time is integrable. When restricted to the class of Hölder functions, this invariant even provides a criterion of cohomology.
We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold.
We study a class of stationary finite state processes, called quasi-Markovian, including in particular the processes whose law is a Gibbs measure as defined by Bowen. We show that, if a factor with integrable coding time of a quasi-Markovian process is maximal in entropy, then this factor splits off, which means that it admits a Bernoulli shift as an independent complement. If it is not maximal in entropy, then we can find a splitting finite extension of this factor, which generalizes a theorem...
We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann–Kakutani odometer), but whose associated Poisson suspension has positive entropy.
We study the generalized random Fibonacci sequences defined by their first non-negative terms and for ≥1,
+2=
+1±
(linear case) and
+2=|
+1±
| (non-linear case), where each ± sign is independent and either + with probability or − with probability 1− (0<≤1). Our main result is that, when is of the form
=2cos(/) for some integer ≥3, the exponential growth of
for 0<≤1,...
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