## Topological Methods in Nonlinear Analysis

### The topological full group of a Cantor minimal system is dense in the full group

#### Abstract

To every homeomorphism $T$ of a Cantor set $X$ one can associate the full group $[T]$ formed by all homeomorphisms $\gamma$ such that $\gamma(x)=T^{n(x)}(x)$, $x\in X$. The topological full group $[[T]]$ consists of all homeomorphisms whose associated orbit cocycle $n(x)$ is continuous. The uniform and weak topologies, $\tau_u$ and $\tau_w$, as well as their intersection $\tau_{uw}$ are studied on ${\rm Homeo}(X)$. It is proved that $[[T]]$ is dense in $[T]$ with respect to $\tau_u$. A Cantor minimal system $(X,T)$ is called saturated if any two clopen sets of "the same measure" are $[[T]]$-equivalent. We describe the class of saturated Cantor minimal systems. In particular, $(X,T)$ is saturated if and only if the closure of $[[T]]$ in $\tau_{uw}$ is $[T]$ and if and only if every infinitesimal function is a $T$-coboundary. These results are based on a description of homeomorphisms from $[[T]]$ related to a given sequence of Kakutani-Rokhlin partitions. It is shown that the offered method works for some symbolic Cantor minimal systems. The tool of Kakutani-Rokhlin partitions is used to characterize $[[T]]$-equivalent clopen sets and the subgroup $[[T]]_x \subset [[T]]$ formed by homeomorphisms preserving the forward orbit of $x$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 16, Number 2 (2000), 371-397.

Dates
First available in Project Euclid: 22 August 2016

https://projecteuclid.org/euclid.tmna/1471875709

Mathematical Reviews number (MathSciNet)
MR1820514

Zentralblatt MATH identifier
0978.37004

#### Citation

Bezuglyi, Sergey; Kwiatkowski, Jan. The topological full group of a Cantor minimal system is dense in the full group. Topol. Methods Nonlinear Anal. 16 (2000), no. 2, 371--397. https://projecteuclid.org/euclid.tmna/1471875709

#### References

• T. Giordano, I. Putnam and C. Skau, Topological orbit equivalence and $C^*$-crossed products , J. Reine Angew. Math., 469 (1995), 51–111 \ref\key 2 ––––, Full groups of Cantor minimal systems, preprint \ref\key 3
• E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems , Internat. J. Math., 6(1995), 569–579 \ref\key 4
• C. Grillenberger, Construction of strictly ergodic systems I. Given entropy , Z. Wahrscheinlichkeitstheorie verw. Geb., 25 (1973), 323–334 \ref\key 5
• R. H. Herman, I. Putnam and C. Skau, Ordered Bratteli diagrams, dimension groups, and topological dynamics , Internat. J. Math., 3(1992), 827–864 \ref\key 6
• K. Jacobs and M. Keane, 0-1 sequences of Toeplitz type , Z. Wahrscheinlichkeitstheorie verw. Geb., 13(1969), 123–131 \ref\key 7
• W. Krieger, On a dimension for a class of homeomorphism groups , Math. Ann., 252(1980), 87–95 \ref\key 8
• N. Ormes, Strong orbit realization for minimal homeomorphisms , J. Anal. Math., 71(1997), 103–133 \ref\key 9
• I. Putnam, The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set , Pacific J. Math., 136(1989), 329–353 \ref\key 10
• S. Williams, Toeplitz minimal flows which are not uniquely ergodic , Z. Wahrscheinlichkeitstheorie verw. Geb., 67(1984), 95–107