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Uncountable ω-limit sets with isolated points

Chris Good, Brian E. Raines, Rolf Suabedissen (2009)

Fundamenta Mathematicae

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....

Uniformly recurrent sequences and minimal Cantor omega-limit sets

Lori Alvin (2015)

Fundamenta Mathematicae

We investigate the structure of kneading sequences that belong to unimodal maps for which the omega-limit set of the turning point is a minimal Cantor set. We define a scheme that can be used to generate uniformly recurrent and regularly recurrent infinite sequences over a finite alphabet. It is then shown that if the kneading sequence of a unimodal map can be generated from one of these schemes, then the omega-limit set of the turning point must be a minimal Cantor set.

Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map

Viktor Avrutin, Ben Futter, Laura Gardini, Michael Schanz (2012)

ESAIM: Proceedings

In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns...

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