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1
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Continuum many tent map inverse limits with homeomorphic postcritical ω-limit sets

100%
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.
2
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Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

81%
Good C., Knight R., Raines B.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 3
267-289
EN
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.
3
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On the ω-limit sets of tent maps

81%
Barwell A., Davies G., Good C.
Fundamenta Mathematicae
|
2012
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tom 217
|
nr 1
35-54
EN
For a continuous map f on a compact metric space (X,d), a set D ⊂ X is internally chain transitive if for every x,y ∈ D and every δ > 0 there is a sequence of points ⟨x = x₀,x₁,...,xₙ = y⟩ such that $d(f(x_i),x_{i+1})< δ$ for 0 ≤ i< n. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.
4
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Shadowing and expansivity in subspaces

81%
Barwell A., Good C., Oprocha P.
Fundamenta Mathematicae
|
2012
|
tom 219
|
nr 3
223-243
EN
We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.
5
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Uncountable ω-limit sets with isolated points

81%
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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