Finite generators of ergodic endomorphisms
Iterations of the Frobenius-Perron operator for parabolic random maps
We describe totally dissipative parabolic extensions of the one-sided Bernoulli shift. For the fractional linear case we obtain conservative and totally dissipative families of extensions. Here, the property of conservativity seems to be extremely unstable.
An Application of Skew Product Maps to Markov Chains
By using the skew product definition of a Markov chain we obtain the following results: (a) Every k-step Markov chain is a quasi-Markovian process. (b) Every piecewise linear map with a Markovian partition defines a Markov chain for every absolutely continuous invariant measure. (c) Satisfying the Chapman-Kolmogorov equation is not sufficient for a process to be quasi-Markovian.
Smooth Extensions of Bernoulli Shifts
For homographic extensions of the one-sided Bernoulli shift we construct σ-finite invariant and ergodic product measures. We apply the above to the description of invariant product probability measures for smooth extensions of one-sided Bernoulli shifts.
Minimal generators for aperiodic endomorphisms
Every aperiodic endomorphism of a nonatomic Lebesgue space which possesses a finite 1-sided generator has a 1-sided generator such that . This is the best estimate for the minimal cardinality of a 1-sided generator. The above result is the generalization of the analogous one for ergodic case.
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