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Inhomogeneities in non-hyperbolic one-dimensional invariant sets

100%
Raines B.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 3
241-268
EN
The topology of one-dimensional invariant sets (attractors) is of great interest. R. F. Williams [20] demonstrated that hyperbolic one-dimensional non-wandering sets can be represented as inverse limits of graphs with bonding maps that satisfy certain strong dynamical properties. These spaces have "homogeneous neighborhoods" in the sense that small open sets are homeomorphic to the product of a Cantor set and an arc. In this paper we examine inverse limits of graphs with more complicated bonding maps. This allows us to understand the topology of a wider class of spaces that includes both hyperbolic and non-hyperbolic attractors. Many of these spaces have the property that most small open sets are homeomorphic to the product of a Cantor set and an arc. The interesting "inhomogeneities" occur away from these neighborhoods. By examining the dynamics of the bonding maps that generate these spaces, we characterize the inhomogeneities, and we show that there is a natural nested hierarchy in the collection of these points that is topological.
2
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Continuum many tent map inverse limits with homeomorphic postcritical ω-limit sets

63%
Good C., Raines B.
Fundamenta Mathematicae
|
2006
|
tom 191
|
nr 1
1-21
EN
We demonstrate that the set of topologically distinct inverse limit spaces of tent maps with a Cantor set for its postcritical ω-limit set has cardinality of the continuum. The set of folding points (i.e. points at which the space is not homeomorphic to the product of a zero-dimensional set and an arc) of each of these spaces is also a Cantor set.
3
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Shadowing and internal chain transitivity

63%
Meddaugh J., Raines B.
Fundamenta Mathematicae
|
2013
|
tom 222
|
nr 3
279-287
EN
The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with shadowing have the property that every internally chain transitive set is an ω-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of ω-limit sets.
4
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Uncountable ω-limit sets with isolated points

51%
Good C., Raines B., Suabedissen R.
Fundamenta Mathematicae
|
2009
|
tom 205
|
nr 2
179-189
EN
We give two examples of tent maps with uncountable (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the ω-limit set is uncountable. Secondly, we give an example of an ω-limit set of the form C ∪ R for which the Cantor set C is minimal.
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