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### Open Access Journals

DCDS

We show that a well known lemma concerning conditions equivalent to topological transitivity is false when posed in a setting that is too general. We also explore some ways of remedying this problem.

keywords:
Topological dynamics.

DCDS

Topological transitivity, weak mixing and non-wandering are definitions used
in topological dynamics to describe the ways in which open sets feed into each other under
iteration. Using finite directed graphs, these definitions are generalized to obtain topological
mapping properties. The extent to which these mapping properties are logically distinct is
examined. There are three distinct properties which entail "interesting" dynamics. Two
of these, transitivity and weak mixing, are already well known. The third does notappear
in the literature but turns out to be close to weak mixing in a sense to be discussed. The
remaining properties comprise a countably infinite collection of distinct properties entailing
somewhat less interesting dynamics and including non-wandering.

DCDS

Spacing subshifts were introduced
by Lau and Zame in 1973 to provide accessible examples of maps that are (topologically) weakly mixing but not mixing. Although they show a rich variety of dynamical characteristics, they have received little subsequent attention in the dynamical systems literature. This paper is a systematic study of their dynamical properties and shows that they may be used to provide examples of dynamical systems with a huge range of interesting dynamical behaviors. In a later paper we propose to consider in more detail the case when spacing subshifts are also sofic and transitive.

DCDS

Spacing shifts were introduced by Lau and Zame in the 1970's to provide accessible examples of maps that are weakly mixing but not mixing. In previous papers by the authors and others, it has been observed that the problem of describing when spacing shifts are topologically transitive appears to be quite difficult in general. In the present paper, we give a characterization of sofic spacing shifts and begin to investigate which sofic spacing shifts are topologically transitive. We show that the canonical graph presentation of such a shift has a rather simple form, for which we introduce the terminology

*hereditary bunched cycle*and discuss the apparently difficult problem of determining which hereditary bunched cycles actually present spacing shifts.## Year of publication

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