On a Bernoulli Property for Multi-dimensional Mappings with Finite Range Structure
Michiko Yuri · Tokyo Journal of Mathematics · 1986
The mapping we consider is characterized by a certain type of parti- tion $Q=\{X_{a}:a\in I\}$ of $X$ and a finite number of subsets $U_{0}(=X),$ $U_{1},$ $\cdots,$ $U_{N}$ of $X$ satisfying some special properties (see \S 1 for precise definitions).We shall call such a transformation $T$ a multi-dimensional mapping with a finite range structure.If such a $T$ satisfies the Renyi's condition, in addition, then it is known that $T$ has a finite absolutely continuous invariant measure, and furthermore, under some additional conditions one can prove that $Q$ is a weak Bernoulli partition ([9], [18]).On the other hand, when $X$ is an interval of $R^{1}$ , Ledrappier established in [6] the weak Bernoulli property for a transformation $T$ having a similar characterization under some further hypothesis, such as the existence of a finite invariant measure with positive entropy, but without assuming that $T$ satisfies the Renyi's condition (cf.[2]).The main ingredient of his proof, which is patterned after the work of Sinai [15] (cf.[16]) and Ratner [12], is the use of Rohlin's formula for proving the absolute continuity of some conditional measures.In this paper we establish a sufficient condition for a multi-dimen- sional mapping with a finite range structure to have the weak Bernoulli property when they do not necessarily satisfy the Renyi's condition.We do need, however, to make several assumptions on the transformation; some of these assumptions seem to be essential, while the others are seen