On the homeomorphism group of the Cantor set: monotone and measure preserving maps
Abstract
Let E be the Cantor set, G=Homeo(E), let P≤ G be the group of order preserving homeomorphisms and L≤ G the group of homeomorphisms preserving the Cantor--Lebesgue measure λ. We study how G is built from these two subgroups. In 2003, Glasner and Weiss stated that G=LP=PL. We show that this is not the case, in fact PL and LP are two different proper subsets of G. On the other hand, PL is a dense Gδ subset of G, the multiplication map P× L→ PL is a homeomorphism, and G=PLP=LPL. In particular, every homeomorphism of E is conjugate to an element of PL. We classify the double cosets PgL using special clopen values set. There are continuum many of them, and PL is the only one which is not meager. We prove that G is topologically generated by one element of P and one element of L and these two generators produce the Thompson's group V. We also show that the generic homeomorphism of E is the product of a generic element of P and a generic element of L. The equality G=PL fails also if λ is replaced by any other full nonatomic measure.
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BibTeX
@misc{vejnar2026homeomorphism,
title = {{On the homeomorphism group of the Cantor set: monotone and measure preserving maps}},
author = {Benjamin Vejnar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32424},
eprint = {2610.32424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32424},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the homeomorphism group of the Cantor set: monotone and measure preserving maps
AU - Benjamin Vejnar
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32424
UR - https://arxiv.org/abs/2610.32424
AB - Let E be the Cantor set, G=Homeo(E), let P≤ G be the group of order preserving homeomorphisms and L≤ G the group of homeomorphisms preserving the Cantor--Lebesgue measure λ. We study how G is built from these two subgroups. In 2003, Glasner and Weiss stated that G=LP=PL. We show that this is not the case, in fact PL and LP are two different proper subsets of G. On the other hand, PL is a dense G_δ subset of G, the multiplication map P× L→ PL is a homeomorphism, and G=PLP=LPL. In particular, every homeomorphism of E is conjugate to an element of PL. We classify the double cosets PgL using special clopen values set. There are continuum many of them, and PL is the only one which is not meager. We prove that G is topologically generated by one element of P and one element of L and these two generators produce the Thompson's group V. We also show that the generic homeomorphism of E is the product of a generic element of P and a generic element of L. The equality G=PL fails also if λ is replaced by any other full nonatomic measure.
ER -