S-limit shadowing is generic for continuous Lebesgue measure-preserving circle maps

Abstract

In this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. In addition, we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.

Periodic points and shadowing for generic Lebesgue measure-preserving interval maps

“…Now we restrict to our particular context. The roots for studying generic properties on C λ (I) come from the paper [10] and this line of study was continued recently in [17], [11], [12], [13]. The first observation we can make about maps from C λ (I) is that they have dense set of periodic points.…”

“…The proof in the circle case follows by combining various known results. First of all the density of a special collection of maps in P A λ (S 1 ) was shown in Lemmas 13 and 14 from [11], we can assume all of the absolute values of slopes of these maps are at least 4. Then using Lemma 12 from [13] we can find a dense set of maps in P A λ (S 1 ) all of whose critical values are distinct and again of whose slopes are at least 4.…”

“…First of all the density of a special collection of maps in P A λ (S 1 ) was shown in Lemmas 13 and 14 from [11], we can assume all of the absolute values of slopes of these maps are at least 4. Then using Lemma 12 from [13] we can find a dense set of maps in P A λ (S 1 ) all of whose critical values are distinct and again of whose slopes are at least 4. Finally using Lemma 12 from [17] whose proof also works without change for the circle we have that there is a dense set of Markov maps in P A λ (S 1 ).…”

“…Returning to the case when M = S 1 , the following result was obtained in [13,Theorem 2]. For each α ∈ [0, 1)], let r α : S 1 → S 1 be a circle rotation for the angle α.…”

Parametrized family of pseudo‐arc attractors: Physical measures and prime end rotations

“…For preliminary results concerning crookedness we adjust in Section 2 techniques developed by Minc and Transue [41] and combine them with a special window perturbations that were first introduced in [18] and subsequently used in [16, 17]. Of central importance in proving Theorem 1.1 is Lemma 2.20, where we show that the Lebesgue measure‐preserving perturbations we construct satisfy certain requirements from [41].…”