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Reading along arithmetic progressions

T. Downarowicz (1999)

Colloquium Mathematicae

Given a 0-1 sequence x in which both letters occur with density 1/2, do there exist arbitrarily long arithmetic progressions along which x reads 010101...? We answer the above negatively by showing that a certain regular triadic Toeplitz sequence does not have this property. On the other hand, we prove that if x is a generalized binary Morse sequence then each block can be read in x along some arithmetic progression.

Reflexively representable but not Hilbert representable compact flows and semitopological semigroups

Michael Megrelishvili (2008)

Colloquium Mathematicae

We show that for many natural topological groups G (including the group ℤ of integers) there exist compact metric G-spaces (cascades for G = ℤ) which are reflexively representable but not Hilbert representable. This answers a question of T. Downarowicz. The proof is based on a classical example of W. Rudin and its generalizations. A~crucial step in the proof is our recent result which states that every weakly almost periodic function on a compact G-flow X comes from a G-representation of X on reflexive...

Relational quotients

Miodrag Sokić (2013)

Fundamenta Mathematicae

Let 𝒦 be a class of finite relational structures. We define ℰ𝒦 to be the class of finite relational structures A such that A/E ∈ 𝒦, where E is an equivalence relation defined on the structure A. Adding arbitrary linear orderings to structures from ℰ𝒦, we get the class 𝒪ℰ𝒦. If we add linear orderings to structures from ℰ𝒦 such that each E-equivalence class is an interval then we get the class 𝒞ℰ[𝒦*]. We provide a list of Fraïssé classes among ℰ𝒦, 𝒪ℰ𝒦 and 𝒞ℰ[𝒦*]. In addition, we classify...

Relatively minimal extensions of topological flows

Mieczysław Mentzen (2000)

Colloquium Mathematicae

The concept of relatively minimal (rel. min.) extensions of topological flows is introduced. Several generalizations of properties of minimal extensions are shown. In particular the following extensions are rel. min.: distal point transitive, inverse limits of rel. min., superpositions of rel. min. Any proximal extension of a flow Y with a dense set of almost periodic (a.p.) points contains a unique subflow which is a relatively minimal extension of Y. All proximal and distal factors of a point...

Remarks on the region of attraction of an isolated invariant set

Konstantin Athanassopoulos (2006)

Colloquium Mathematicae

We study the complexity of the flow in the region of attraction of an isolated invariant set. More precisely, we define the instablity depth, which is an ordinal and measures how far an isolated invariant set is from being asymptotically stable within its region of attraction. We provide upper and lower bounds of the instability depth in certain cases.

Rotation sets for graph maps of degree 1

Lluís Alsedà, Sylvie Ruette (2008)

Annales de l’institut Fourier

For a continuous map on a topological graph containing a loop S it is possible to define the degree (with respect to the loop S ) and, for a map of degree 1 , rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop S then the set of rotation numbers of points in S has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational...

Rotation sets for subshifts of finite type

Krystyna Ziemian (1995)

Fundamenta Mathematicae

For a dynamical system (X,f) and a function φ : X N the rotation set is defined. The case when (X,f) is a transitive subshift of finite type and φ depends on the cylinders of length 2 is studied. Then the rotation set is a convex polyhedron. The rotation vectors of periodic points are dense in the rotation set. Every interior point of the rotation set is a rotation vector of an ergodic measure.

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