# On connectedness and dimension of a Besicovitch space over SZ.

Abstract : We prove that the topological space szd, proposed in is path-connected and has infinite dimension. The latter property makes of this space a more natural setting for cellular automata when they are considered as a solutions of difference equations. In fact, difference equations are defined on an infinite dimensional space. On the contrary the classical product topology on SZ is zero-dimensional. Moreover we present a transitive dynamical system on szd, whose existence was given as an open problem in. Another interesting property that we prove is that szd is not separable. This property partially explain the difficulty'' of finding transitive systems on such a space. We also prove that some properties of Toeplitz sequences on szd and as a byproduct we obtain a weak fixed point'' theorem for continuous mappings on szd. Finally we sketch an interesting connection between infinite Sturmian words and szd.
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https://hal-lara.archives-ouvertes.fr/hal-02102095
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Submitted on : Wednesday, April 17, 2019 - 9:14:07 AM
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• HAL Id : hal-02102095, version 1

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Enrico Formenti, Petr Kurka. On connectedness and dimension of a Besicovitch space over SZ.. [Research Report] LIP RR-1998-03, Laboratoire de l'informatique du parallélisme. 1998, 2+12p. ⟨hal-02102095⟩

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