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Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

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Good C., Knight R., Raines B.

Fundamenta Mathematicae

2006

tom 192

nr 3

267-289

We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or attracting condition. We show that for a family of conceptual models for the Hénon attractor, inverse limits of tent maps, these restrictions characterize the structure of inhomogeneities. We end with several results regarding the collection of parameters that generate such spaces.

Set-theoretic constructions of two-point sets

100%

Chad B., Knight R., Suabedissen R.

Fundamenta Mathematicae

2009

tom 203

nr 2

179-189

A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Knight R.

1 Chad B.

1 Good C.

1 Raines B.

1 Suabedissen R.

1 2009

1 2006

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Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

Set-theoretic constructions of two-point sets