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Acta Mathematica Hungarica
Authors: Shigeki Akiyama, Tibor Borbély, Horst Brunotte, Attila Pethő, and Jörg M. Thuswaldner

Summary We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

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In the following text we prove that in a generalized shift dynamical system (X Г, σ φ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent:

  1. the dynamical system (X Г, σ φ) is chaotic in the sense of Devaney
  2. the dynamical system (X Г, σ φ) is topologically transitive
  3. the map φ: Г → Г is one to one without any periodic point.
Also for infinite countable Г and finite discrete X with at least two elements (X Г, σ φ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

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