Abstract:

Among all notions of chaos, there are three widely accepted: Devaney chaos, LiYorke chaos and (positive) topological entropy. It is known that exact Devaney chaos, i.e. an exact map with dense set of periodic points, satisfies all these three notions. Various results establish the existence of maps with properties related to chaos (e.g., transitivity) for specific spaces such as the interval, the Cantor set or the Lelek fan, as well as for broader classes, including manifolds and dendrites. Furthermore, chaotic behavior often emerges as a generic phenomenon in the sense of Baire category. Together with Benjamin Vejnar, we prove that every Peano continuum (i.e. a locally connected continuum) admits exact Devaney chaos. Additionally, we generalize some prior results by showing that if a Peano continuum X satisfies the condition that selfmaps locally constant on some dense open subset form a dense subset of all selfmaps, then:

• exactly Devaney chaotic maps form a dense subset of chain transitive selfmaps of X,

• mixing is generic among chain transitive self-maps of X,

• shadowing is generic among all self-maps of X.