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

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.


Introduction
The notion of shadowing (or pseudo orbit tracing property, see Definition 6) is a classical notion in the theory of dynamical systems. It was defined as a tool for better understanding of asymptotic aspects of diffeomorphisms dynamics independently by Anosov [1] and Bowen [7]. Informally, the shadowing property ensures that computational errors do not accumulate in the following sense: in the systems with shadowing property the approximate trajectories will reflect real dynamics up to some small error that is made in each iteration. In particular, this is of great importance in systems with sensitive dependence on initial conditions, where small errors may potentially result in large divergence of trajectories.
While we are still lacking the full classification of systems with shadowing, there are classes where its occurrence has been completely characterized. To look only at the most general results, all uniformly hyperbolic systems have shadowing property and Walters characterized the symbolic dynamical systems with shadowing property, they are shifts of finite type (see the books [22,24] for more explanation). A useful collection of conditions characterizing shadowing in the latter setting was recently provided by Good and Meddaugh [9].
We call a property generic if it is satisfied on at least a dense G δ of the underlying space. A naturally related question which attracted attention of many researchers is the genericity of shadowing in dynamical systems. Hyperbolic systems are known to be rather special, and finding an answer in other classes of functions usually turns out to be a delicate matter. The first results in this direction were obtained in dimension one by Yano in [25] for the space of homeomorphisms on the unit circle and Odani in [21] for all smooth manifolds of the dimension at most 3. A particularly nice technique was introduced by Pilyugin and Plamenevskaya in [23] who proved genericity of shadowing for homeomorphisms on any smooth compact manifold without boundary. This result was later extended using topological tools to a wider context (e.g. see [13]).
Mizera proved [19] that shadowing is a generic property in the class of continuous maps of the interval or circle. Recently, these results were extended to many other one-dimensional spaces, see [11,14,20]. It turned out that non-invertibility is not an obstacle to obtain genericity of shadowing also in higher dimension [15].
In the literature there are many different generalizations of the shadowing property. Among the most natural ones is the limit shadowing property (see Definition 6), which was introduced by Pilyugin at al. [8]. In this definition, error in consecutive elements of pseudo-trajectories tends to zero (so-called asymptotic pseudo orbit), but we require that accuracy of tracing increases with time. While limit shadowing seems completely different than shadowing, it was proved in [12] that transitive maps with limit shadowing also have the shadowing property. Recently [2], it was proven that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-sphere satisfy both the shadowing and the limit shadowing property.
In general, it can happen that for an asymptotic pseudo orbit which is also a δ-pseudo orbit, the point which ε-traces it and the point which traces it in the limit are two different points [3]. This shows that possessing a common point for such a tracing is a stronger property than the shadowing and limit shadowing properties together. The described property was introduced in [17] and is called s-limit shadowing property (again see Definition 6 for the precise definition). Not much is known about s-limit shadowing or even limit shadowing with respect to genericity in particular classes of functions. Besides the results mentioned above, the only result known to the authors which barely touches this problem is [18], where it is proven that in the class of continuous maps on manifolds of dimension m ≥ 1, s-limit shadowing is a dense property with respect to the metric of uniform convergence.
The main difficulty in proving denseness or genericity of s-limit shadowing is its "instability"; meaning that, intuitively, arbitrarily small perturbations can destroy it. Therefore, even the density result in [18] relies on a very careful control of consecutive perturbations. Our main theorem here, in particular, addresses the following very general question from [18]: (Q2) Is s-limit shadowing a C 0 -generic property on spaces where shadowing is generic?
Let λ denote the normalized Lebesgue measure on I := [0, 1] and letλ denote the normalized Lebesgue measure on S 1 . The particular setting that we are interested in this paper is the family of continuous Lebesgue measure preserving maps of the unit circle Cλ(S 1 ) endowed with topology of uniform convergence, which makes it a complete space. Topological and measure theoretical properties of generic Lebesgue measure preserving interval maps were studied in [4,6,5]. We obtain the following two new results: Theorem 1. The s-limit shadowing property is generic in Cλ(S 1 ). Corollary 2. The limit shadowing, periodic shadowing and shadowing property are generic in Cλ(S 1 ).
In the context of Lebesgue measure preserving functions, the genericity of shadowing was recently proven in [10] for homeomorphisms on manifolds (with or without boundary) of dimension at least 2 where the authors use Oxtoby-Ulam's theorem [16] and its underlying subdivision of any such manifold. For manifolds of dimension 1 it is natural to ask analogous questions for non-invertible maps and results here can also be viewed as a contribution along this line of research. Let C λ (I) denote the family of Lebesgue measure preserving maps equipped with the metric of uniform convergence. For C λ (I), the genericity of shadowing and periodic shadowing was proven recently in [5]; therefore, the results obtained here can be viewed as strengthening of those results. However, we need to note that proving the genericity of s-limit shadowing turns out to be very delicate and, in particular, we cannot apply the main idea of the proof to the interval setting (see the explanation in Section 4).
Our result (with simplifications of the proof) holds in an even looser environment. By C(S 1 ) we denote the class of continuous interval maps endowed with the topology of uniform convergence. In this setting we obtain the following two new results.
Theorem 3. The s-limit shadowing property is generic in C(S 1 ). Corollary 4. The limit shadowing property is generic in C(S 1 ).
Let us outline the structure of the paper. In Preliminaries we first review the definitions related to the shadowing property that we address in our context of Lebesgue measure preserving circle maps. Then we review the basic setting of C λ (S 1 ) which we work with in the rest of the paper. We start Section 3 by outlining the proof of main theorem. In Subsection 3.2 we restrict our attention to particular families of maps in C λ (S 1 ) and we study their properties; we use these families and their properties later in the proof of s-limit shadowing. In Subsection 3.3 the proof of s-limit shadowing starts. We pose five conditions (C1)-(C5) that our partitions and special perturbations need to satisfy. In the rest of this section we address how to get such partitions and perturbations from machinery developed in Subsection 3.2. Subsection 3.4 gives the proof of Theorem 1 using the assumptions given by conditions (C1)-(C5) in Subsection 3.3. We conclude the paper with Section 4 where we give a brief explanation why the proof of s-limit shadowing as presented in this paper can not work in the setting of Lebesgue measure preserving interval maps.
2.1. Shadowing property. First we give the definition of shadowing property and its related extensions that we use in this paper.
Definition 5. For δ > 0 and a map f ∈ C(S 1 ) we say that a sequence of points If a sequence {x k } k∈N0 ⊂ S 1 is a δ-pseudo orbit and an asymptotic pseudo-orbit then we say that it is an asymptotic δ-pseudo orbit. Definition 6. We say that a map f ∈ C(S 1 ) has the: • shadowing property if for every ε > 0 there exists δ > 0 satisfying the following condition: given a δ-pseudo orbit y = {y n } n∈N0 we can find a corresponding point x ∈ S 1 which ε-traces y, i.e., d(f n (x), y n ) < ε for every n ∈ N 0 .
• periodic shadowing property if for every ε > 0 there exists δ > 0 satisfying the following condition: given a periodic δ-pseudo orbit y = {y n } n∈N0 we can find a corresponding periodic point x ∈ S 1 , which ε-traces y. • limit shadowing if for every asymptotic pseudo orbit {x n } n∈N0 ⊂ S 1 there exists p ∈ S 1 such that d(f n (p), x n ) → 0 as n → ∞.
• s-limit shadowing if for every ε > 0 there exists δ > 0 so that (1) for every δ-pseudo orbit y = {y n } n∈N0 we can find a corresponding point x ∈ S 1 which ε-traces y, (2) for every asymptotic δ-pseudo orbit y = {y n } n∈N0 of f , there is x ∈ S 1 which ε-traces y and lim n→∞ d(y n , f n (x)) = 0.
Remark 7. Note that s-limit shadowing implies both classical and limit shadowing.

2.2.
Lebesgue measure preserving circle maps. Consider a continuous map f : LetF : R → R be a lifting of f , i.e., the continuous map for which Note that since two liftings of f differ by a integer constant, F does not depend on a concrete choice of a lifting of f . In what follows the set of all liftings, resp. representatives of onto circle maps will be denotedF(R), resp. F([0, 1)).

Remark 8. One can easily see that a circle map f is onto if and only if its repre
Letλ denote the normalized Lebesgue measure on S 1 and B the Borel sets in S 1 . In this paper we will work with continuous maps from S 1 into S 1 preserving the measureλ, which we denote We consider the set Cλ(S 1 ) equipped with the uniform metric ρ: We leave the standard proof of the following fact to the reader.
The following conditions are equivalent. ( Proof. Let us assume thatF is a lifting of f and denote ψ = φ|[0, 1). Then ψ is a continuous bijection. From (1) we get Moreover, A ⊂ [0, 1) andÃ := ψ(A) ⊂ S 1 are simultaneously Borel and Assuming (i), using (2) and (3) we can write This shows that the statements (i) implies (ii). If (ii) is true we can writẽ We say that a map from F([0, 1)) is piecewise affine if it has finitely many affine pieces of monotonicity. We will say thatF ∈F(R) is piecewise affine if its corresponding representativeF |[0, 1)) is piecewise affine. In general, maps from F([0, 1)) are not continuous but they can be piecewise monotone and smooth or even piecewise affine. For these cases the following lemma states a useful criterium about when an element F of F([0, 1)) represents f ∈ Cλ(S 1 ).
Lemma 11. Let F ∈ F([0, 1)) be a piecewise affine representative with nonzero slopes and such that its derivative does not exist at a finite set E. Then the properties (i) and (ii) from Lemma 10 are equivalent to the property Proof. By the hypothesis the set F (E) is finite and for each y thus Lemma 10(ii) implies (4).

The proof
3.1. Outline of the proof. The proof of our main result, Theorem 1, relies on four rather technical steps. The first step is treated in Lemmas 12, 13 and 14 and consists of the construction of a special dense subset Cλ ,0 (S 1 ) of Cλ(S 1 ). Let Q π := Q + π. The maps in Cλ ,0 (S 1 ) are piecewise affine and every map g from Cλ ,0 (S 1 ) fulfills the key property In particular, the maps in Cλ ,0 (S 1 ) have all points of discontinuity of derivatives in φ(Q π ) so the equation (6) applies. In the second step in Lemmas 16 and 17 we twice perturb maps from Cλ ,0 (S 1 ) to obtain maps satisfying the list of conditions (C1)-(C5) from Subsection 3.3. Applying in Subsection 3.4 both perturbations and also the result of Lemma 18 on a sequence {g m } m≥1 dense in Cλ ,0 (S 1 ), we arrive to new sequences {θ m } m≥1 of maps from Cλ(S 1 ), their neighborhoods {U m } m≥1 and also carefully constructed partitions {Q m } m≥1 . In particular, using (6) we can ensure that for some pairs m < n, Q n is a refinement of Q m . The final step consists of the proof that all maps in have the s-limit shadowing property.
3.2. Particular families of Lebesgue measure preserving circle maps and its representatives. In this subsection we will define particular families of Lebesgue measure preserving circle maps that we will apply later for the construction of partitions needed for the proof of genericity of s-limit shadowing property in our context. For a piecewise affine mapF ∈F(R) (i.e.F |[0, 1)) is piecewise affine) we denote by T (F ), resp. D(F ) the turning points, resp. the set of points of discontinuity of derivative ofF . We also put Let us recall our convention that is stated before Remark 8; the setF(R) consists of liftings of onto circle maps. LetF 0 (R) ⊂F(R) be defined as (7) Since the set Q π is dense in R, we have the following lemma.
Proof. FixF ∈F(R) and ε > 0. Clearly there exists a piecewise affine map F ∈F(R) with nonzero slopes such that Notice that for a piecewise affineF the set (D(F ) ∩ [0, 1]) \ F −1 (S) has to be either empty or finite. For any x ∈ (D(F ) ∩ [0, 1]) \ F −1 (S) we can proceed in two steps. First, we modify the graph of F on a small neighbourhood of x as it is shown in Figure 1. Second, denoting the new maps F ,F , arrange both new turning points and also all preimages of their images to be from D(F ) ∩ Q π and therefore reduce the number Repeating the described modification finitely many times, we fulfill (7)(ii), i.e., F ∈F 0 (R).
Consider a liftingF ∈F 0 (R) introduced by (7) and the corresponding representative F =F |[0, 1)(mod 1). Define the outer homeomorphism h : [0, 1] → [0, 1] by (8) h(0) = 0 and h(x) = λ(F −1 ((0, x))), x ∈ (0, 1]. The picture on the right represents a Lebesgue measure preserving map G, however the lifting of this map is not from the setF 0 since the mapsF and F do not have their turning points (black squares) and also preimages of images of turning points that are not turning points (black discs) in Q π .
Clearly, by (7) and Remark 8 the map F is surjective with nonzero slopes, h is an increasing continuous piecewise affine function satisfying h(0) = 0 and h(1) = 1. In particular, h is a homeomorphism of [0, 1]. The set of all liftings of maps from Cλ(S 1 ) will be denoted byF λ (R).
Lemma 13. LetF ∈F 0 (R) be lifting of f ∈ C(S 1 ), F its corresponding representative and h defined as in (8). For the map G = h • F the following is true.   Figure 3. Let r ∈ Q and let α = π − r > 0 be a small irrational number. On the left picture, graph of functionF represents a shift (i.e. rotation on the circle for the original circle map) of the representative F from Figure 2 for α to the right (and its lift, similarly as in Figure 2). Due to the choice of α, the liftingF (x+α) will already be fromF 0 (R). Note that the outer homeomorphism forF stays the same as the one in Figure 2.
(v) This is because the slopes of piecewise affine outer homeomorphism h can change only at the points from F (T (F )).
(vi) For each interval (u, v), where (1) either u = 0 and v is the least value F (x) > 0 at a turning point x ofF , (2) or u, v are two consecutive values at turning points ofF , (3) or u is the biggest value F (x) < 1 at a turning point x ofF and v = 1, F −1 ((u, v)) can be expressed as a finite union It follows from our definition ofF 0 (R) in (7) that a j , b j ∈ Q π for all j, so Fix a turning point w ∈ T [0,1] (F ) for which F (w) > 0. One can set where u i , v i were described above in (1)-(3); then by (8) and (9), By (v) and (7) (10) and (7)(ii) we obtainG(D(G)) =G(D(F )) ⊂ Q hencẽ (vii) This property is a consequence of (iv) and the fact thatG is a lifting of g due to formula (1).
Recall that by our definition the setF(R) consists of liftings of onto circle maps (see Remark 8 and the text preceding it). Lemma 14. The set Cλ ,0 (S 1 ) is dense in Cλ(S 1 ).
Proof. Fix ε > 0 and a map e ∈ Cλ(S 1 ) with a liftingẼ ∈F λ (R). By Lemma 12 and Remark 8 there is a mapF ∈F 0 (R) such that its representative F =F |[0, 1) is onto and 1 ((0, x))) (as in (8)) Condition (ii) can be fulfilled due to the following reasoning. If circle maps f n converge in the uniform metric to a Lebesgue measure preserving circle map then their corresponding h n defined as in (ii) converge to id -we refer the reader to [4] where the analogous interval case had been treated in details. We have proved in Lemma 13(iv) that G = h • F is a representative of a map g from Cλ ,0 (S 1 ).
Definition 15. We say that two maps f, g :
3.3. Partitions, special perturbations. In this section we will start with maps from Cλ ,0 (S 1 ) defined in the previous section and particular associated partitions of S 1 and show how to perturb such maps and refine their associated partitions so that they will satisfy conditions (C1)-(C6) given below. This will provide us with the crucial step in proving genericity of s-limit shadowing in the next section. For what follows we refer the reader to see Figure 5 to visualize the discussed concepts better. Given a piecewise affine circle map g ∈ Cλ ,0 (S 1 ), ε > 0 and its affine partition P ⊃ D(g) for which P ⊂ φ(Q π ) and ||P|| < ε, (where || · || denotes the maximum diameter of partition elements) we will construct a perturbation θ of g and a partition Q ⊂ φ(Q π ) for θ for which P ≺ Q (i.e. Q refines P) and such that each J ∈ Q has a subdivision into subarcs L J 1 , L J 2 , M J , R J 2 , R J 1 whose order preserves order in J satisfies (C1) There is I ∈ Q (depending on J) such that θ(J) ⊃ I, ) ⊂ θ(J) for sufficiently small η > 0. By (11), the map g has its liftingG from Fλ ,0 (R) represented by G = h • F ∈ F([0, 1)), where F and h were described immediately prior to Lemma 13. By (iii),(iv) of Lemma 13, g is piecewise affine, i.e., such that the mapG, resp. G is piecewise affine. Applying (8) and Lemma 13 we can consider a finite set P of points such that for whichG|[p i , p i+1 ] is affine for each i (set p m+1 = p 1 + 1), and for We will call the set P , resp. P a partition forG, resp. g. RedefiningG on each [p i , p i+1 ] by (the numbers n(i) ∈ N and vectorx(i) will be specified later) where β's were introduced in (12) and s(i) ∈ {+, −} ares chosen to satisfyΣ i (p i ) = G(p i ), yields a mapΣ : Notice that stillΣ(1) −Σ(0) = deg(g), so abusing the notation we will again denote byΣ its extension from [0, 1] to the whole real line keeping the ruleΣ(x+1) =Σ(x)+deg(g). In fact the mapΣ is a lifting of some map σ : S 1 → S 1 . Because by Lemma 16 each mapG|[p i , p i+1 ] has been replaced by a λ-equivalent mapΣ i , it follows that the map Σ ∈ F([0, 1)) representing σ satisfies the conditions of Lemma 11 hence by Lemma 10 (i) it holds that σ ∈ Cλ(S 1 ). For the mapΣ|[0, 1] we will consider a new partition for some m ′ ∈ N, where the vectorsx(i) = (x 0 (i), x 1 (i), . . . , x 2n(i)+1 (i)) will be chosen to satisfy Q ⊂ Q π . Thus, the set Q contains P and also all new turning points ofΣ in (0, 1) being in Q \ P . From our specific choice of β's in (16) and Lemma 13(viii) we obtain Q ⊂ Q π andΣ(Q) ⊂ Q; denoting Q = φ(Q) we analogously obtain for σ and Q Q ⊂ φ(Q π ) and σ(Q) ⊂ φ(Q); which implies that σ(Q) ∩ Q = ∅. At the same time the numbers n(i) (recall that the number of full laps of β is 2n(i)+1) can be taken sufficiently large to satisfy for each i and arcs Up to now, using rescaled versions of β's we have perturbated the mapG (resp. g) on the intervals [p i , p i+1 ] (resp. arcs φ([p i , p i+1 ])) to obtain the liftingΣ of σ ∈ Cλ(S 1 ).
In the last part of this proof we will proceed similarly: using rescaled versions of Ψ's from Figure 4 we will perturb the mapΣ (resp. σ) on the intervals [q i , q i+1 ] (resp. arcs φ([q i , q i+1 ])) to obtain the liftingΘ of θ ∈ Cλ(S 1 ).
Therefore, for each i define The reason why the degree preserving extension ofΘ to the real line is a lifting of a map θ ∈ Cλ(S 1 ) is analogous as above: the map θ is represented by the map Θ =Θ|[0, 1)(mod 1) ∈ F([0, 1)) that fulfills conditions of Lemma 11. Let us consider the mapΘ, resp. θ with respect to partition Q, resp. Q. For what follows we refer the reader to the right picture in Figure 4. Taking in (17) , ε i and d i sufficiently close to 0 and e i sufficiently close to 1, with the help of (18) we can ensure that for each i, ; then using φ we can transfer these sets to the arc . The sketch of this construction is drawn in Figure 5.
Proof. Let x = {x s } ∞ s=0 be a δ-pseudo orbit for τ . We claim that there is a sequence of arcs J s ∈ Q = Q ε,θ and sets Q s ⊂ J s such that (1) x s ∈ J s , (2) τ (Q s ) ⊃ Q s+1 and Q s ∈ {L Js 1 , R Js 1 }.
As J 0 ∈ Q select any arc such that x 0 ∈ J 0 (in the worst case there are two such arcs). Fix any Q 0 ∈ {L J0 1 , R J0 1 }. Now suppose that the above conditions are satisfied for some s and let J s+1 ∈ Q be such that x s+1 ∈ J s+1 . If θ(x s ) ∈ J s+1 then since θ(J s ) contains at least one element of Q, by condition (C1) we have that θ(J s )∩L in respective cases and observe that the claim holds. But then, since ||Q|| < ε, it is enough to choose z ∈ ∩τ −s (Q s ) to obtain a point ε-tracing x.
In the previous section we have described a special type of perturbation of a piecewise affine map g ∈ Cλ ,0 (S 1 ) ⊂ Cλ(S 1 ) resulting with a circle map θ. The main property of θ was stated in Lemma 18. Now we are going to apply similar approach to a dense sequence of piecewise affine maps from Cλ ,0 (S 1 ) which is possible by invoking Lemma 14. To that end, let Γ := {g m } m≥1 ⊂ Cλ ,0 (S 1 ) be a dense sequence of maps in Cλ(S 1 ) such that • each g m has an affine partition P m ⊂ φ(Q π ) satisfying ||P m || < 1 m , • for each n ≥ m, g n (P m ) ∩ P m = ∅.
Notice that the second property is guaranteed by Lemma 13(viii). Following the previous section we perturb g m to θ m with ε = 1 m , corresponding partition Q m := Q 1 m ,θm , η m , δ m = δ(θ m ) < 1 m and U m an open neighborhood around θ m in Cλ(S 1 ) such that • the boundary of U m does not intersect Γ.
We will proceed as follows to construct sequences {Q m } ∞ m=1 , where each Q m is a subset of φ(Q π ), and {U m } ∞ m=1 : we repeatedly use Lemma 13(viii) and Lemma 14 (1) we perturb g 1 to θ 1 to obtain g n (Q 1 ) ∩ Q 1 = ∅ for each n ≥ 1, (2) for m > 1, having already constructed the sets Q i and U i , i = 1, . . . , m − 1, in order to construct Q m and U m we distinguish two possibilities: (a) either g m / ∈ m−1 i=1 U i and then we construct Q m and U m to fulfill is the largest number with this property; denoting E(Q i ) the set of φ-images of points defined in (19) for all In particular, Q m is an affine partition for θ m which is a refinement of Q i and g n (Q m ) ∩ Q m = ∅ for each n ≥ m. In addition we require that the boundary of U m does not intersect Γ; this is possible since Γ is countable.
Let us put A n = m≥n U m . Clearly, each A n is open and dense so the intersection Proof of Theorem 1. We will prove that each τ ∈ A has the s-limit shadowing property. By our definition of A, there is an increasing sequence {m(k)} ∞ k=1 such that be an asymptotic pseudo orbit. By Lemma 18 the partition Q m(i) and δ = δ m(i) > 0 were chosen for α-shadowing with α = 1/m(i). Fix k so that (20) 4/k < δ.
Assume for simplicity that x is a δ-pseudo orbit and it is a γ-pseudo orbit for all s ≥ N − 1 for some N . Let J s ∈ Q m(i) and Q s ⊂ J s be provided as in (1),(2) of the proof of Lemma 18 for x and by the same conditions, let R s ∈ Q m(j) , and let W s ⊂ R s for s ≥ N − 1 be provided by the fact that τ ∈ U m(j) ⊂ B δ m(j) (θ m(j) ). In particular, First, if W N ⊂ τ (Q N −1 ) then we can switch directly from the arc Q N −1 used for α-tracing to the arc W N used for β-tracing.
which gives a contradiction. On the other hand, x N −1 ∈ J N −1 and d(x N , τ (x N −1 )) < γ and diam(R N ) < 1/k. Also W N ⊂ R N and x N ∈ R N . Then if ξ := γ + 1/k, where the last inclusion is a consequence of (21). Since W N is not included in the arc τ (Q N −1 ) and both diameters diam L JN 2 and diam R JN 2 are greater than 2δ, the only possibility is that W N ⊂ L JN 2 ∪ M JN ∪ R JN 2 . But by (C5) we have B 4δ (θ m(i) (W N )) ⊂ θ m(i) (J N ) = θ m(i) (Q N ), and thus since τ ∈ U m(i) , B 3δ (θ m(i) (W N )) ⊂ τ (Q N ) and (23) B 2δ (τ (W N )) ⊂ τ (Q N ).
Gluing (23) and (24) together, we get This allows us to switch from the arc Q N used for α-tracing to the arc W N +1 used for β-tracing. Then using inductively the above construction we obtain that for every ε > 0, every τ ∈ n≥1 A n and every asymptotic pseudo orbit x = {x s } ∞ s=0 which is δpseudo orbit, we can find a sequence of closed arcs I s ⊂ S 1 with the following properties: (1) τ (I s ) ⊃ I s+1 , (2) diam(I s ∪ {x s }) < ε, (3) for every β > 0 there is N > 0 such that diam(I s ∪ {x s }) < β for all s > N .
Now it is enough to take any z ∈ s≥0 τ −s (I s ) to ε-trace and asymptotically trace x.

Final remarks
As we mentioned before, some inspiration for this paper comes from [18] where it is proved that on manifolds (including dimension one) s-limit shadowing is dense in the class of continuous maps. In particular, it is dense in continuous maps on the circle and the interval. It was also proven in our recent paper [5] that s-limit shadowing is dense also in Lebesgue measure preserving maps on the interval. Then, in the view of the above results and the results in the present paper it is natural to expect that s-limit shadowing is generic also in Lebesgue measure preserving interval maps. Unfortunately, the proof of Theorem 1 will not directly work in that case as we explain below. The main technique in our proof is showing that W N ⊂ τ (Q N −1 ) or W N +1 ⊂ τ (Q N ) under the map τ which is small perturbation of θ, see the discussion after (22), for more details. While for small perturbation we may ensure that τ (Q N −1 ) ⊃ Q N , we cannot control covering of smaller sets W s by Q s , see Figure 6 for an intuitive explanation of possible problems. This situation may happen near endpoints of the interval, where we cannot guarantee sufficiently long overlapping of τ (Q N −1 ) or τ (Q N ). Such a situation does not happen on the circle due to the lack of boundary. This motivates us to state the following question.
Question A. Is s-limit shadowing generic in Lebesgue measure preserving maps on the interval?  Figure 6. After perturbation the image of Q N −1 = Q N covers itself. Therefore, ε-tracing is still possible, however the image of Q N −1 = Q N does not cover W N = W N +1 anymore.
As we explained above, possible positive answer to the above question will require some new techniques, beyond the ones used in the present work. On the other hand, a standard technique to disprove that a condition is generic is to find an open set without the property. Such approach is again impossible, because we have proven [5] that s-limit shadowing is dense in C λ (S 1 ).