Distributional Chaos and Dendrites

Abstract

Many definitions of chaos have appeared in the last decades and with them the question if they are equivalent in some more specific spaces. Our focus will be distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. In the end of the paper we will also briefly look at the correlation with other types of chaos.

Dynamics on dendrites with closed endpoint sets

“…This extends the work of Kočan, Kurková, and Málek, who use the Gehman dendrite and its subdendrites to represent every shiftinvariant subset of {0, 1} N -in this way, any subshift can be embedded into a dendrite map so that all other points are eventually fixed [8]. Shift spaces are a rich source of examples and counterexamples in dynamical systems, and this construction has led to an abundance of dendrite maps with various combinations of chaotic and non-chaotic behavior [3,4,7,9,13]. Our construction allows us to start with other zero-dimensional systems and gives us isometric embeddings rather than just topological ones.…”

Ultragraph shift spaces and chaos

“…We start this section recalling the three versions of distributional chaos (as found in [30]). Then, in Proposition 3.11, we show the equivalence between chaos in an ultragraph shift space and the existence of a vertex which is the base of two distinct closed paths.…”

“…We remark that although we are only working with ultragraph shift spaces, distributional chaos can be defined for any dynamical system over a metric space (with the same definition). For instance, in [26], Schweizer and Smítal introduced distributional chaos in the context of continuous maps of the interval, and later this definition was split into three versions of distributional chaos (briefly, DC1, DC2, and DC3), as we can see in [27] and more recently in [30]. Moreover, a subset S of X is distributionally scrambled of type i (or a DCi set), where i = 1, 2, 3, if every pair of distinct elements in S is a DCi pair.…”

Some Chaos Notions on Dendrites

“…The interest in the study of chaotic behaviour on dendrites has been increasing for a few years. For examples, researches focus on shadowing [17], distributional chaos [18], induced map [19], and relations between some chaotic notions [20][21][22][23]. It turns out that most of the results on the interval do not necessarily hold true on dendrites.…”