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On Rigid Minimal Spaces.

Jan P Boroński1,2, Jernej Činč1,2, Magdalena Foryś-Krawiec1,2

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Summary
This summary is machine-generated.

This study explores rigid minimal spaces, revealing distinct classes of spaces that admit minimal homeomorphisms but not minimal maps. It also demonstrates the existence of minimal spaces with degenerate homeomorphism groups.

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Area of Science:

  • Topology
  • Dynamical Systems
  • Set Theory

Background:

  • Minimal spaces are defined by maps whose orbits are dense.
  • Research is motivated by studies on cyclic homeomorphism groups and powers of minimal homeomorphisms.

Purpose of the Study:

  • To investigate the relationship between spaces with cyclic homeomorphism groups and spaces where homeomorphism powers are related.
  • To introduce new classes of minimal spaces and explore their properties.

Main Methods:

  • Comparing properties of spaces with cyclic homeomorphism groups generated by minimal homeomorphisms.
  • Analyzing spaces where the square of every homeomorphism is a power of a minimal homeomorphism.
  • Constructing examples of minimal spaces with degenerate homeomorphism groups.

Main Results:

  • Demonstrated that spaces with cyclic homeomorphism groups and spaces with specific homeomorphism power properties do not coincide.
  • Introduced a new class of spaces admitting minimal homeomorphisms but lacking minimal maps.
  • Provided the first examples of minimal spaces with degenerate homeomorphism groups.
  • Developed a method for constructing decomposable compact, connected spaces with cyclic homeomorphism groups generated by a minimal homeomorphism.

Conclusions:

  • The two classes of spaces considered are distinct.
  • New examples of minimal spaces with unique homeomorphism group properties have been established.
  • A method to construct specific types of minimal spaces has been provided, addressing an open question.