A transitive homeomorphism on the Lelek fan

Transitive points in CR-dynamical systems

“…Suppose that f is surjective. The implication from 1 to 2 follows from [8,Theorem 2.15]. To prove the implication from 2 to 1, suppose that (lim ← − − (X, f ), σ −1 ) is transitive.…”

“…The Lelek fan and the Cantor fan are very different but on the other hand, they share many properties, for example, they both admit transitive homeomorphisms [3,8], each one of them embeds into another and they are both universal for the class of smooth fans [5]. Therefore, it is only natural, to expect that also the Lelek fan might admit similar chaotic properties as does the Cantor fan.…”

“…Before that, we give an easy proof (independently from the results about quotients of dynamical systems) that for the Lelek fan L, there is a homeomorphism (as well as a continuous function which is not a homeomorphism) f : L → L such hat (L, f ) is chaotic in the sense of Robinson but not in the sense of Devaney. To do this, we first recall the basic construction from [8] We also define the set M r,ρ by M r,ρ = X + L r,ρ and we use σ r,ρ to denote the shift map σ + L r,ρ on M r,ρ . In the following theorem, we prove that the dynamical system from Theorem 5.3 also has sensitive dependence on initial conditions.…”