Generic chaos on graphs

Generic chaos on dendrites

“…To prove that generic chaos together with these two conditions imply generic ε-chaos, we first study invariant subcontinua for a generically chaotic map on a dendrite satisfying the two conditions. Similarly as on the interval [55] and graphs [59], we are able to prove that the invariant non-degenerate subdendrites have large diameters; see Lemma 4.3. Of course, the topology of dendrites makes the proof much more complicated, among other reasons due to the phenomenon described on p. 2127.…”

“…We know from [55] that the interval is such a space. By [59], even graphs are such spaces. However, in general, this is not true for dendrites; indeed, by [40], an ω-star admits a generically chaotic selfmap that is not generically ε-chaotic for any ε > 0 (an ω-star is a (topologically unique) dendrite having exactly one branch point, and this branch point is of infinite order).…”

Generic chaos on dendrites

“…To prove that generic chaos together with these two conditions imply generic ε-chaos, we first study invariant subcontinua for a generically chaotic map on a dendrite satisfying the two conditions. Similarly as on the interval [55] and graphs [59], we are able to prove that the invariant nondegenerate subdendrites have large diameters, see Lemma 4.3. Of course, the topology of dendrites makes the proof much more complicated, among other reasons due to the phenomenon described in Footnote 6.…”

“…We now consider one-dimensional spaces. By [59], X contains all finite graphs and, by our Theorem A, it contains many, but not all, dendrites. We conjecture that Theorem A can be extended to local dendrites, i.e., we conjecture that a local dendrite is in X if and only if it is completely regular, with all points of finite order.…”

“…However, the following is true. Lemma 2.14 (Proposition 13 in [59]). Let X be a compact metric space, f : X → X be continuous and Y be a regular closed f -invariant subset of X.…”