Properties of Shadowable Points: Chaos and Equicontinuity

Shadowable Point for Non-Autonomous Discrete Dynamical Systems

“…A shadowable point of a continuous map is defined to be a point such that the shadowing lemma holds for pseudo-orbits beginning at the point. After that, Kawaguchi extends the study on shadowable points recently introduced by Morales in relation to chaotic or non-chaotic properties [4]. About the non-autonomous case, Duarte and Klein give the shadowing property for nonautonomous systems which satisfy several conditions of the maps f n and pseudo-orbits in Avalanche principle proof [5].…”

Positive entropy through pointwise dynamics

“…We start stating a result due to Kawaguchi, N. which exhibits an abstraction of the techniques developed in [13] by Moothathu and Oprocha. Lemma 6 (Lemma 2.3 in [5]). Let f : X → X be a continuous map and let…”

“…Thus for any two distinct sequences s 1 , s 2 ∈ Σ, points which b-shadow γ(s 1 ) and γ(s 2 ) must be distinct. If we now define Y ⊂ X as the set of all the b-shadows x γ(s) for the δ-orbits γ(s), then we can define a surjection π : Y → Σ setting π(x γ(s) ) = s. For more details about the proof of the previous lemmas, see [5].…”

Topological stability and shadowing of zero-dimensional dynamical systems

“…Our proof of Theorem 1.2 and Proposition 1.1 relies on the decompositions of X into the equivalence classes with respect to the chain relations, which we briefly describe below. We remark that such arguments have been already used in several papers (see [9] for details).…”

“…Put D i = [f i (p)], 0 ≤ i ≤ m − 1, and D m = D 0 . Then, as shown in [9], we have the following properties.…”