The simplest erasing substitution

Abstract

In this work, we begin the study of a new class of dynamical systems determined by interval maps generated by the symbolic action of erasing substitution rules. We do this by discussing in some detail the geometric, analytical, dynamical and arithmetic properties of a particular example, which has the virtue of being arguably the simplest and that at the same time produces interesting properties and new challenging problems.

Mixing properties of erasing interval maps

“…The present paper extends the results of [5, 6] by widening the perspective from purely topological to measurable dynamical properties, specifically mixing.…”

“…Some properties of this map were already investigated in [6], where it was denoted by R . We point out that the map R was a slightly different object than .…”

“…For the reader convenience, we recall here the main results of the two aforementioned papers, namely [5, 6], because we are now ready to formulate them more precisely: in [6] it has been proven that is Devaney chaotic, exhibits distributional chaos of type (in the sense of [1]), has infinite topological entropy and uncountably many periodic cycles for every order; moreover, it is Borel-singular and not bi-measurable, because there are sets of Lebesgue measure zero whose -image is not measurable; in [5], Li–Yorke and Devaney chaos, as well as infinite topological entropy, have been established for maps generated by a more general class of erasing substitutions. …”

“…in [6] it has been proven that is Devaney chaotic, exhibits distributional chaos of type (in the sense of [1]), has infinite topological entropy and uncountably many periodic cycles for every order; moreover, it is Borel-singular and not bi-measurable, because there are sets of Lebesgue measure zero whose -image is not measurable;…”

“…Some works within theoretical computer science have addressed the problem of extending classical results on substitutions to the erasing case (see, for instance, [10, 11, 20]). On a more specifically dynamical ground, the study of maps generated by the action of erasing substitutions on the binary expansion of reals was begun in [5, 6]. In our opinion, the main interest of these systems arises from the combination of very simple defining rules with rich properties and dynamical behavior.…”