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Intrinsic ergodicity of affine maps in \([0,1]^ d\). (English) Zbl 0886.58024
A well-studied class of maps of the interval consists of the so-called \(\beta\)-transformations. For a \(\beta>1\), such a transformation sends \(x\) to the fractional part of \(\beta x\). In spite of being discontinuous (even as circle maps), \(\beta\)-transformations have good ergodic properties.
The present paper proves many of the same results for an analogous class of maps of the \(d\)-dimensional unit cube into itself. If an expanding affine map \(A\) is given, a transformation of this type sends \(x\) to the vector of fractional parts of \(Ax\). It is proved that these maps have only finitely many ergodic invariant probability measures of maximal entropy, only one if the system is topologically transitive, and that among invariant probability measures those of maximal entropy, not necessarily invariant, are exactly the absolutely continuous ones. The main technique of the paper is the use of Hofbauer’s Markov diagrams.

MSC:
37E99 Low-dimensional dynamical systems
37A99 Ergodic theory
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