The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury¹ ; Daniel J. Rudolph²

¹ Department of Mathematics University of North Texas Denton, TX 76203, U.S.A.
² Department of Mathematics Colorado State University Fort Collins, CO 80523, U.S.A.

Fundamenta Mathematicae, Tome 198 (2008) no. 2, pp. 149-163.

Résumé

Two invertible dynamical systems $(X,\mathfrak{F}A,\mu ,T)$ and $(Y,\mathfrak{F}B,\nu ,S)$, where $X$ and $Y$ are Polish spaces and Borel probability spaces and $T$, $S$ are measure preserving homeomorphisms of $X$ and $Y$, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping $\varphi$ from a subset $X_0$ of $X$ of measure one onto a subset $Y_0$ of $Y$ of full measure such that (1) $\varphi {|}_{X_0}$ is continuous in the relative topology on $X_0$ and ${\varphi }^{-1}{|}_{Y_0}$ is continuous in the relative topology on $Y_0$, (2) $\varphi ({\mathrm{O}\mathrm{r}\mathrm{b}}_T(x))={\mathrm{O}\mathrm{r}\mathrm{b}}_S(\varphi (x))$ for $\mu$-a.e. $x\in X$.$(X,\mathfrak{F}A,\mu ,T)$ and $(Y,\mathfrak{F}B,\nu ,S)$ are said to be finitarily evenly Kakutani equivalent if they are finitarily orbit equivalent by a mapping $\varphi$ for which there are measurable subsets $A$ of $X$ and $B=\varphi (A)$ of $Y$ with $\varphi$ an isomorphism of $T_A$ and $T_B$.It is shown here that the Morse minimal system and the binary odometer are finitarily evenly Kakutani equivalent.

Détail

DOI : 10.4064/fm198-2-5

Mots-clés : invertible dynamical systems mathfrak mathfrak where polish spaces borel probability spaces measure preserving homeomorphisms said finitarily orbit equivalent there exists invertible measure preserving mapping phi subset measure subset full measure phi continuous relative topology phi continuous relative topology phi orb orb phi mu a mathfrak mathfrak said finitarily evenly kakutani equivalent finitarily orbit equivalent mapping phi which there measurable subsets phi phi isomorphism nbsp shown here morse minimal system binary odometer finitarily evenly kakutani equivalent