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For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...
We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples...
A control system is said to be finite if the Lie algebra generated by its vector fields
is finite dimensional. Sufficient conditions for such a system on a compact manifold to be
controllable are stated in terms of its Lie algebra. The proofs make use of the
equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010)
956–973]. and of the existence of an invariant measure on certain compact homogeneous
spaces.
A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.
A control system is said to be finite if the Lie algebra generated by its vector fields
is finite dimensional. Sufficient conditions for such a system on a compact manifold to be
controllable are stated in terms of its Lie algebra. The proofs make use of the
equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010)
956–973]. and of the existence of an invariant measure on certain compact homogeneous
spaces.
We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by Lasota and Yorke (1982). Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations on...
We consider a family of transformations with a random parameter and study a random dynamical system in which one transformation is randomly selected from the family and applied on each iteration. The parameter space may be of cardinality continuum. Further, the selection of the transformation need not be independent of the position in the state space. We show the existence of absolutely continuous invariant measures for random maps on an interval under some conditions.
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