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Adding machines, endpoints, and inverse limit spaces

Lori Alvin, Karen Brucks (2010)

Fundamenta Mathematicae

Let f be a unimodal map in the logistic or symmetric tent family whose restriction to the omega limit set of the turning point is topologically conjugate to an adding machine. A combinatoric characterization is provided for endpoints of the inverse limit space (I,f), where I denotes the core of the map.

Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows

T. Downarowicz, Y. Lacroix (1998)

Studia Mathematica

Let ( Z , T Z ) be a minimal non-periodic flow which is either symbolic or strictly ergodic. Any topological extension of ( Z , T Z ) is Borel isomorphic to an almost 1-1 extension of ( Z , T Z ) . Moreover, this isomorphism preserves the affine-topological structure of the invariant measures. The above extends a theorem of Furstenberg-Weiss (1989). As an application we prove that any measure-preserving transformation which admits infinitely many rational eigenvalues is measure-theoretically isomorphic to a strictly ergodic toeplitz...

Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker (2014)

Fundamenta Mathematicae

In this paper we consider those Fraïssé classes which admit companion classes in the sense of [KPT]. We find a necessary and sufficient condition for the automorphism group of the Fraïssé limit to be amenable and apply it to prove the non-amenability of the automorphism groups of the directed graph S(3) and the boron tree structure T. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering G L ( V ∞ ) , where V ∞ is the countably infinite-dimensional...

An attraction result and an index theorem for continuous flows on ℝ n × [ 0 , ∞ )

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on E = ℝ n + 1 for which ∂ E = ℝ n × 0 is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ℝ n × [ 0 , ∞ ) .

An exotic flow on a compact surface

N. Markley, M. Vanderschoot (2000)

Colloquium Mathematicae

In 1988 Anosov [1] published the construction of an example of a flow (continuous real action) on a cylinder or annulus with a phase portrait strikingly different from our normal experience. It contains orbits whose ο m e g a -limit sets contain a non-periodic orbit along with a simple closed curve of fixed points, but these orbits do not wrap down on this simple closed curve in the usual way. In this paper we modify some of Anosov’s methods to construct a flow on a surface of genus 2 with equally striking...

Attractors and Inverse Limits.

James Keesling (2008)

RACSAM

This paper surveys some recent results concerning inverse limits of tent maps. The survey concentrates on Ingram’s Conjecture. Some motivation is given for the study of such inverse limits.

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