We consider skew products $T(x,y)=(f\left(x\right),{T}_{e\left(x\right)}y)$ preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and ${T}_{i}$, i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these...

Let (f,α) be the process given by an endomorphism f and by a finite partition $\alpha ={{A}_{i}}_{i=1}^{s}$ of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if $E(f,\alpha )\subset g:{\bigvee}_{{{B}_{i}}_{i=1}^{s}}suppg={\bigcup}_{i=1}^{s}{A}_{i}\times {B}_{i}$. We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the ’chequer-wise’ partition for σ × S, where σ is...

We consider the skew product transformation T(x,y)= (f(x), ${T}_{e\left(x\right)}$) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and ${{T}_{s}}_{s\in S}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling...

We give an elementary proof for the uniqueness of absolutely continuous invariant measures for expanding random dynamical systems and study their mixing properties.

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.

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.

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.

Every aperiodic endomorphism $f$ of a nonatomic Lebesgue space which possesses a finite 1-sided generator has a 1-sided generator $\beta $ such that ${k}_{f}\le card\phantom{\rule{0.166667em}{0ex}}\beta \le {k}_{f}+1$. 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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