We show that a certain type of conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also consider distribution asymptotics of information; e.g. for Boole’s transformation, information is asymptotically mod-normal, a property shared by certain ergodic, probability preserving transformations with zero entropy. §0 Intr...

In this note we prove some results concerning volume-preserving Anosov diffeomorphisms on compact manifolds. The main theorem is that if a C, α > 0, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. This is not necessarily true for C maps. The proof uses a special type of measure density points different from the ...

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...

In this paper, the abstract computational principles underlying topographic maps are discussed. We give a definition of a “perfectly neighbourhood preserving” map, which we call a topographic homeomorphism, and we prove that this has certain desirable properties. It is argued that when a topographic homeomorphism does not exist (the usual case), many equally valid choices are available for quan...

It is well known that if G is a countable amenable group and G y (Y ,ν) factors onto G y (X ,μ), then the entropy of the first action must be at least the entropy of the second action. In particular, if G y (X ,μ) has infinite entropy, then the action G y (Y ,ν) does not admit any finite generating partition. On the other hand, we prove that if G is a countable nonamenable group then there exis...

The Monge mass transfer problem, as proposed by Monge in 1781, is to move points from one mass distribution to another so that a cost functional is minimized among all measure preserving maps. The existence of an optimal mapping was proved by Sudakov in 1979, using probability theory. A proof based on partial differential equations was recently found by Evans andGangbo. In this paperwegive amor...

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing new any factor map between topological dynamical systems, concept conditional entropy. It is proved Ledrappier's type variational principle concerning Consequently, if has zero entropy (such called asymptotically $ h $-expansive), then there exi...