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Minimal fixed point sets of relative maps

Xue Zhao (1999)

Fundamenta Mathematicae

Let f: (X,A) → (X,A) be a self map of a pair of compact polyhedra. It is known that f has at least N(f;X,A) fixed points on X. We give a sufficient and necessary condition for a finite set P (|P| = N(f;X,A)) to be the fixed point set of a map in the relative homotopy class of the given map f. As an application, a new lower bound for the number of fixed points of f on Cl(X-A) is given.

Minimal nonhomogeneous continua

Henk Bruin, Sergiǐ Kolyada, L'ubomír Snoha (2003)

Colloquium Mathematicae

We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.

Minimal non-invertible transformations of solenoids

Dariusz Tywoniuk (2012)

Colloquium Mathematicae

We construct a continuous non-invertible minimal transformation of an arbitrary solenoid. Since solenoids, as all other compact monothetic groups, also admit minimal homeomorphisms, our result allows one to classify solenoids among continua admitting both invertible and non-invertible continuous minimal maps.

Minimal self-joinings and positive topological entropy II

François Blanchard, Jan Kwiatkowski (1998)

Studia Mathematica

An effective construction of positive-entropy almost one-to-one topological extensions of the Chacón flow is given. These extensions have the property of almost minimal power joinings. For each possible value of entropy there are uncountably many pairwise non-conjugate such extensions.

Mixing properties of nearly maximal entropy measures for ℤ d shifts of finite type

E. Robinson, Ayşe Şahin (2000)

Colloquium Mathematicae

We prove that for a certain class of ℤ d shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

Module-valued functors preserving the covering dimension

Jan Spěvák (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove a general theorem about preservation of the covering dimension dim by certain covariant functors that implies, among others, the following concrete results. If G G is a pathwise connected separable metric...

More on ordinals in topological groups

Aleksander V. Arhangel'skii, Raushan Z. Buzyakova (2008)

Commentationes Mathematicae Universitatis Carolinae

Let τ be an uncountable regular cardinal and G a T 1 topological group. We prove the following statements: (1) If τ is homeomorphic to a closed subspace of G , G is Abelian, and the order of every non-neutral element of G is greater than 5 then τ × τ embeds in G as a closed subspace. (2) If G is Abelian, algebraically generated by τ ⊂ G , and the order of every element does not exceed 3 then τ × τ is not embeddable in G . (3) There exists an Abelian topological group H such that ω 1 is homeomorphic to a closed subspace...

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