New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

被引:1
作者
Boronski, Jan P. [1 ,2 ]
Clark, Alex [3 ]
Oprocha, Piotr [1 ,2 ]
机构
[1] AGH Univ Sci & Technol, Fac Appl Math, Al Mickiewicza 30, PL-30059 Krakow, Poland
[2] Inst Res & Applicat Fuzzy Modeling, Natl Supercomp Ctr IT4Innovat, Div Univ Ostrava, 30 Dubna 22, Ostrava 70103, Czech Republic
[3] Univ Leicester, Dept Math, Mile End Rd, London E1 4NS, England
关键词
Pseudo-circle; Topological entropy; Minimal set; Weak mixing; Uniformly rigid; Hereditaritly indecomposable continuum; ROTATION SETS; ENTROPY; DIFFEOMORPHISMS; CONSTRUCTION; MAPS; CLASSIFICATION; HOMEOMORPHISM; PSEUDOCIRCLES; PROPERTY; SPACES;
D O I
10.1007/s10884-021-10069-3
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel-Anosov-Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.
引用
收藏
页码:1175 / 1201
页数:27
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