No CrossRef data available.
Article contents
ORIENTATION OF PIECEWISE POWERS OF A MINIMAL HOMEOMORPHISM
Published online by Cambridge University Press: 15 November 2021
Abstract
We show that, given a compact minimal system
$(X,g)$
and an element h of the topological full group
$\tau [g]$
of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of
$(X,G)$
into minimal and periodic parts, where G is any virtually polycyclic subgroup of
$\tau [g]$
. We also use the orientation of orbits to give a refinement of the index map and to describe the role in
$\tau [g]$
of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that
$h \in \tau [g]$
.
MSC classification
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 113 , Issue 2 , October 2022 , pp. 226 - 256
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by Anthony Henderson
Research funded by ARC project FL170100032.
References
Belinskaya, R. M., ‘Partitions of Lebesgue space in trajectories defined by ergodic automorphisms’, Funct. Anal. Appl. 2(3) (1968), 190–199.CrossRefGoogle Scholar
Cornulier, Y., ‘Groupes plein-topologiques (d’après Matui, Juschenko, Monod, …)’, Astérisque 361(1064, VIII) (2014), 183–223.Google Scholar
Garrido, A. and Reid, C. D., ‘Discrete locally finite full groups of Cantor space homeomorphisms’, Bull. Lond. Math. Soc. 53(4) (2021), 1228–1248.CrossRefGoogle Scholar
Giordano, T., Putnam, I. F. and Skau, C. F., ‘Topological orbit equivalence and
${C}^{\ast }$
-crossed products’, J. reine angew. Math. 469 (1995), 51–112.Google Scholar
${C}^{\ast }$
-crossed products’, J. reine angew. Math. 469 (1995), 51–112.Google ScholarGiordano, T., Putnam, I. F. and Skau, C. F., ‘Full groups of Cantor minimal systems’, Israel J. Math. 111(1) (1999), 285–320.CrossRefGoogle Scholar
Herman, R. H., Putnam, I. F. and Skau, C. F., ‘Ordered Bratteli diagrams, dimension groups and topological dynamics’, Int. J. Math. 3(6) (1992), 827–864.CrossRefGoogle Scholar
Juschenko, K. and Monod, N., ‘Cantor systems, piecewise translations and simple amenable groups’, Ann. of Math. (2) 178(2) (2013), 775–787.CrossRefGoogle Scholar
Krieger, W., ‘On a dimension for a class of homeomorphism groups’, Math. Ann. 252 (1980), 87–95.CrossRefGoogle Scholar
Le Maître, F., ‘On a measurable analogue of small topological full groups’, Adv. Math. 332(9) (2018), 235–286.CrossRefGoogle Scholar
Matui, H., ‘Topological full groups of étale groupoids’, in: Operator Algebras and Applications, Abel Symposia, 12 (eds. Carlsen, T. M., Larsen, N. S., Neshveyev, S. and Skau, C.) (Springer, Cham, 2016), Ch. 10, 203–230.CrossRefGoogle Scholar
Petersen, K. and Shapiro, L., ‘Induced flows’, Trans. Amer. Math. Soc. 177 (1973), 375–390.CrossRefGoogle Scholar
Putnam, I. F., ‘The
${C}^{\ast }$
-algebras associated with minimal homeomorphisms of the Cantor space’, Pacific J. Math. 136(2) (1989), 329–353.CrossRefGoogle Scholar
${C}^{\ast }$
-algebras associated with minimal homeomorphisms of the Cantor space’, Pacific J. Math. 136(2) (1989), 329–353.CrossRefGoogle ScholarReid, C. D., ‘Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces’, Groups Geom. Dyn. 14(2) (2020), 413–425.CrossRefGoogle Scholar
Rubin, M., ‘On the reconstruction of topological spaces from their groups of homeomorphisms’, Trans. Amer. Math. Soc. 312(2) (1989), 487–538.CrossRefGoogle Scholar
Štěpánek, P. and Rubin, M., ‘Homogeneous Boolean algebras’, in: Handbook of Boolean Algebras, Volume 2 (eds. J. D. Monk with R. Bonnet) (1989), Ch. 18, 679–715.Google Scholar
Terada, T., ‘Spaces whose all nonempty clopen subspaces are homeomorphic’, Yokohama Math. J. 40(2) (1993), 87–93.Google Scholar
Tomiyama, J., ‘Topological full groups and structure of normalizers in transformation group
${C}^{\ast }$
-algebras’, Pacific J. Math. 173(2) (1996), 571–583.CrossRefGoogle Scholar
${C}^{\ast }$
-algebras’, Pacific J. Math. 173(2) (1996), 571–583.CrossRefGoogle Scholar