Fractal, group theoretic, and relational structures on Cantor space

Cantor sets;Fractals;Group theory;Topological spaces

Donoven, Casey Ryall ; Falconer, K. J. ; Falconer, K. J.

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Cantor sets;    Fractals;    Group theory;    Topological spaces;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/11370/CaseyDonovenPhDThesis.pdf?sequence=2&isAllowed=y

来源: DR-NTU

【 摘 要 】

Cantor space, the set of infinite words over a finite alphabet, is a type of metric spacewith a `self-similar' structure. This thesis explores three areas concerning Cantor spacewith regard to fractal geometry, group theory, and topology.We find first results on the dimension of intersections of fractal sets within the Cantorspace. More specifically, we examine the intersection of a subset E of the n-ary Cantorspace, C[sub]n with the image of another subset Funder a random isometry. We obtainalmost sure upper bounds for the Hausdorff and upper box-counting dimensions of theintersection, and a lower bound for the essential supremum of the Hausdorff dimension.We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of theCantor space built from transducers. These groups can be seen as homeomorphismsthat respect the self-similar and symmetric structure of C[sub]n, and are supergroups of theHigman-Thompson groups V[sub]n. We explore their isomorphism classes with our primaryresult being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is asemiregular subgroup of the symmetric group on n points.Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing thesequotients by invariant relations as an inverse limit, before characterizing a specific classof fractals known as Sierpiński relatives as invariant factors. We then compare relationsarising through edge replacement systems to invariant relations, detailing the conditionsunder which they are the same.