2.1 Shadowing property

First we give the definition of the shadowing property and its related extensions that we use in this paper.

2.2 Lebesgue measure-preserving circle maps

Consider a continuous map $f\colon ~\mathbb S^1\to \mathbb S^1$ of degree $\mathrm {deg}(f)\in \mathbb {Z}$ . Let $\tilde F\colon ~\mathbb {R}\to \mathbb {R}$ be a lifting of f, that is, the continuous map for which

(1) $$ \begin{align} \phi\circ \tilde F = f\circ\phi\quad\text{on }\mathbb{R}, \end{align} $$

where $\phi \colon ~\mathbb {R}\to \mathbb {S}^1$ is defined by $\phi (x)= e^{2\pi ix}$ . Then $\tilde F(x + 1) = \tilde F(x) + \mathrm {deg}(f)$ for each $x\in \mathbb {R}$ . If $F=\tilde F\vert [0,1)(\mathrm {mod}~ 1)$ , we say that $F\colon ~[0,1)\to [0,1)$ represents f. Note that since two liftings of f differ by an integer constant, F does not depend on a concrete choice of a lifting of f.

In what follows the set of all liftings (respectively, representatives) of onto circle maps will be denoted $\tilde {{\mathcal {F}}}(\mathbb {R})$ (respectively, ${{\mathcal {F}}}([0,1))$ ).

Let $\tilde \lambda $ denote the normalized Lebesgue measure on $\mathbb {S}^1$ and ${\mathcal B}$ the Borel sets in $\mathbb {S}^1$ . In this paper we will work with continuous maps from $\mathbb {S}^1$ into $\mathbb {S}^1$ preserving the measure $\tilde \lambda $ , which we denote by

$$ \begin{align*}C_{\tilde\lambda}(\mathbb{S}^1)=\{f\colon~\mathbb{S}^1\to \mathbb{S}^1\colon\text{for all } A\in{\mathcal B},~\tilde\lambda(A)=\tilde\lambda(f^{-1}(A))\}.\end{align*} $$

We consider the set $C_{\tilde \lambda }(\mathbb {S}^1)$ equipped with the uniform metric $\rho $ :

$$ \begin{align*}\rho (f,g) := \mathop{\mathrm{sup}}\limits_{x \in \mathbb{S}^1} |f(x) - g(x)|.\end{align*} $$

We leave the standard proof of the following fact to the reader.

The next lemma describes elements of ${\mathcal {F}}([0,1))$ representing maps from $C_{\tilde \lambda }(\mathbb {S}^1)$ . We denote by $\lambda $ the Lebesgue measure on $[0,1)$ .

Proof. Let us assume that $\tilde F$ is a lifting of f and denote $\psi =\phi \vert [0,1)$ . Then $\psi $ is a continuous bijection. From equation (1) we get

(2) $$ \begin{align}\psi\circ F = f\circ\psi\quad\text{ on }[0,1). \end{align} $$

Moreover, $A\subset [0,1)$ and $\tilde A:=\psi (A)\subset \mathbb {S}^1$ are simultaneously Borel and

(3) $$ \begin{align}\lambda(A)=\tilde\lambda(\tilde A). \end{align} $$

Assuming (i), using (2) and (3) we can write

$$ \begin{align*}\lambda(A)=\lambda(\psi^{-1}(\tilde A))=\lambda(\psi^{-1}(f^{-1}(\tilde A)))=\lambda(F^{-1}(\psi^{-1}(\tilde A)))=\lambda(F^{-1}(A)). \end{align*} $$

This shows that the statement (i) implies (ii). If (ii) is true we can write

$$ \begin{align*}&\tilde\lambda(\tilde A)=\lambda(A)=\lambda(F^{-1}(A))=\lambda(F^{-1}(\psi^{-1}(\tilde A)))=\lambda(\psi^{-1}(f^{-1}(\tilde A)))=\tilde\lambda(f^{-1}(\tilde A)), \end{align*} $$

so (ii) implies (i).

We say that a map from ${\mathcal {F}}([0,1))$ is piecewise affine if it has finitely many affine pieces of monotonicity. We will say that $\tilde F\in \tilde {\mathcal {F}}(\mathbb {R})$ is piecewise affine if its corresponding representative $\tilde F\vert [0,1))$ is piecewise affine. In general, maps from ${\mathcal {F}}([0,1))$ are not continuous but they can be piecewise monotone and smooth or even piecewise affine. For these cases the following lemma states a useful criterion about when an element F of ${\mathcal {F}}([0,1))$ represents $f\in C_{\tilde \lambda }(\mathbb {S}^1)$ .

Lemma 2.7 Let $F\in {\mathcal {F}}([0,1))$ be a piecewise affine representative with non-zero slopes and such that its derivative does not exist at a finite set E. Then properties (i) and (ii) from Lemma 2.6 are equivalent to the property

(4) $$ \begin{align} \text{for all}~y\in [0,1)\setminus F(E)\colon~\sum_{x\in F^{-1}(y)}\frac{1}{\vert F'(x)\vert}=1. \end{align} $$

Proof. By the hypothesis the set $F(E)$ is finite and for each $y\in (0,1)\setminus F(E)$ we can write, for $J(y,\varepsilon )=[y-\varepsilon ,y+\varepsilon ]$ ,

$$ \begin{align*} \lim_{\varepsilon\to 0_+}\sum_{K\in\mathrm{Comp}(F^{-1}(J(y,\varepsilon)))}\frac{\lambda(K)}{\lambda(J(y,\varepsilon))}=\sum_{x\in F^{-1}(y)}\frac{1}{\vert F'(x)\vert}, \end{align*} $$

thus Lemma 2.6(ii) implies (4).

For the other direction, assuming that property (ii) is not true, one can find some closed interval $J_0\subset [0,1)\setminus F(E)$ and $\delta>0$ for which

(5) $$ \begin{align}\frac{\lambda(J_0)}{\lambda(F^{-1}(J_0))}\in (0,1-\delta)\cup (1+\delta,\infty).\end{align} $$

Then, proceeding inductively, we can detect a nested sequence $J_0\supset J_1\supset \cdots $ of closed intervals fulfilling (5) and $\bigcap _{i=0}^{\infty } J_i=\{y\}$ with $y\in (0,1)\setminus F(E)$ . By (5), equation (4) fails in such y. This shows that (4) implies Lemma 2.6(ii).

3.1 Outline of the proof

The proof of our main result, Theorem 1.1, relies on four rather technical steps. The first step is treated in Lemmas 3.1, 3.2 and 3.3 and consists of the construction of a special dense subset $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ of $C_{\tilde \lambda }(\mathbb {S}^1)$ . Let $\mathbb {Q}_{\pi } := \mathbb {Q} + \pi $ . The maps in $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ are piecewise affine and every map g from $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ fulfills the key property

(6) $$ \begin{align}g(\phi(\mathbb{Q}_{\pi}))\subset \phi(\mathbb{Q}).\end{align} $$

In particular, the maps in $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ have all points of discontinuity of derivatives in $\phi (\mathbb {Q}_{\pi })$ , so equation (6) applies. In the second step in Lemmas 3.5 and 3.6 we twice perturb maps from $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ to obtain maps satisfying the list of conditions (C1)–(C5) from §3.3. Applying in §3.4 both perturbations and also the result of Lemma 3.7 on a sequence $\{g_m\}_{m\ge 1}$ dense in $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ , we arrive at new sequences $\{\theta _m\}_{m\ge 1}$ of maps from $C_{\tilde \lambda }(\mathbb {S}^1)$ , their neighbourhoods $\{U_m\}_{m\ge 1}$ and also carefully constructed partitions $\{\mathcal {Q}_m\}_{m\ge 1}$ . In particular, using (6) we can ensure that for some pairs $m<n$ , $\mathcal {Q}_n$ is a refinement of $\mathcal {Q}_m$ . The final step consists of the proof that all maps in

$$ \begin{align*}A=\bigcap_{n\ge 1}\bigcup_{m\ge n}U_m\end{align*} $$

have the s-limit shadowing property.

3.2 Particular families of Lebesgue measure-preserving circle maps and their 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 map $\tilde F\in \tilde {\mathcal {F}}(\mathbb {R})$ (that is, $\tilde F\vert [0,1))$ is piecewise affine) we denote by $T(\tilde F)$ (respectively, $D(\tilde F)$ ) the turning points (respectively, the set of points of discontinuity of derivative) of $\tilde F$ . We also put $T_{[0,1]}(\tilde F)=T(\tilde F)\cap [0,1]$ and denote

$$ \begin{align*}\mathbb{Q}_{\pi}=\{r+\pi\colon r\in\mathbb{Q}\}. \end{align*} $$

Let us recall our convention that is stated before Remark 2.4; the set $\tilde {\mathcal {F}}(\mathbb {R})$ consists of liftings of onto circle maps. Let $\tilde {\mathcal {F}}_0(\mathbb {R})\subset \tilde {\mathcal {F}}(\mathbb {R})$ be defined as

(7) $$ \begin{align} \tilde F\in\tilde{\mathcal{F}}_0(\mathbb{R})~\equiv~ \begin{cases} \text{(i)}~\tilde F\vert[0,1)\text{ is piecewise affine with non-zero slopes},\\ \text{(ii) for}~S=\{0\}\cup F(T_{[0,1]}(\tilde F)),~D(\tilde F)\cap [0,1]= F^{-1}(S)\subset \mathbb{Q}_{\pi}. \end{cases} \end{align} $$

Since the set $\mathbb {Q}_{\pi }$ is dense in $\mathbb {R}$ , we have the following lemma.

Proof. Fix $\hat F\in \tilde {\mathcal {F}}(\mathbb {R})$ and $\varepsilon>0$ . Clearly there exists a piecewise affine map $\tilde F\in \tilde {\mathcal {F}}(\mathbb {R})$ with non-zero slopes such that:

• • $\mathop {\mathrm {sup}}_{x\in [0,1]}\vert \hat F(x)-\tilde F(x)\vert <\varepsilon $ ;

• • $\hat F(1)-\hat F(0)=\tilde F(1)-\tilde F(0)$ ;

• • $T(\tilde F)\subset \mathbb {Q}_{\pi }$ ;

• • for $F=\tilde F\vert [0,1)(\mathrm {mod}~ 1)$ , if $x\in (0,1)$ satisfies $F(x)\in S$ then $x\in D(\tilde F)\cap \mathbb {Q}_{\pi }$ , so $(D(\tilde F)\cap [0,1])\supset F^{-1}(S)$ .

Notice that for a piecewise affine $\tilde F$ the set $(D(\tilde F)\cap [0,1])\setminus F^{-1}(S)$ has to be either empty or finite. For any $x\in (D(\tilde F)\cap [0,1])\setminus F^{-1}(S)$ we can proceed in two steps. First, we modify the graph of F on a small neighbourhood of x as shown in Figure 1. Second, denoting the new maps by F, $\tilde F$ , arrange both new turning points and also all preimages of their images to be from $D(\tilde F)\cap \mathbb {Q}_{\pi }$ and therefore reduce the number

$$ \begin{align*} \#(D((\tilde F)\cap [0,1])\setminus F^{-1}(S)). \end{align*} $$

Repeating the described modification finitely many times, we fulfil (7)(ii), that is, $\tilde F\in \tilde {\mathcal {F}}_0(\mathbb {R})$ .

Figure 1 Adjustments from the proof of Lemma 3.1.

Consider a lifting $\tilde F\in \tilde F_0(\mathbb {R})$ introduced by (7) and the corresponding representative $F=\tilde F\vert [0,1)(\mathrm {mod}~1)$ . Define the outer homeomorphism $h:[0,1]\to [0,1]$ by

(8) $$ \begin{align} h(0)=0\quad\text{and}\quad h(x)=\lambda(F^{-1}((0,x))),~x\in (0,1]. \end{align} $$

Clearly, by (7) and Remark 2.4 the map F is surjective with non-zero 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 $C_{\tilde \lambda }(\mathbb {S}^1)$ will be denoted by $\tilde {\mathcal {F}}_{\lambda }(\mathbb {R})$ .

For the visual representation of the notions from Lemma 3.2, we refer the reader to Figures 2 and 3.

Figure 2 In this figure the numbers along the graph lines represent slopes of respective affinity pieces. The upper left-hand graph shows a lifting of a non-Lebesgue measure-preserving circle map $\tilde F$ restricted on $[0,1)$ (taking into account the dashed lines) and also of F, its corresponding representative (without the dashed lines). The lower left-hand graph depicts the corresponding outer homeomorphism h. The right-hand graph represents a Lebesgue measure-preserving map G; however, the lifting of this map is not from the set $\tilde {{\mathcal {F}}}_0$ since the maps $\tilde F$ 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 $\mathbb {Q}_{\pi }$ .

Figure 3 Let $r\in \mathbb {Q}$ and let $\alpha =\pi -r>0$ be a small irrational number. The left-hand graph of function $\hat F$ represents a shift (that is, rotation on the circle for the original circle map) of the representative F from Figure 2 for $\alpha $ to the right (and its lift, similarly to Figure 2). Due to the choice of $\alpha $ , the lifting $\tilde F(x+\alpha )$ will already be from $\tilde {\mathcal {F}}_0(\mathbb {R})$ . Note that the outer homeomorphism for $\hat F$ stays the same as the one in Figure 2.

Proof. To verify (i), for $0\le u<v\le 1$ we can write, with the help of (8),

$$ \begin{align*} &\lambda(G^{-1}((u,v)))=\lambda(F^{-1}(h^{-1}(u,v)))=\lambda(F^{-1}((h^{-1}(u),h^{-1}(v))))\\ &\quad =\lambda(F^{-1}((0,h^{-1}(v))))-\lambda(F^{-1}((0,h^{-1}(u))))=v-u. \end{align*} $$

(ii) This is because $h(x)=0$ if and only if $x=0$ .

(iii) This follows from the fact that $G=h\circ F$ and the outer map h is an increasing continuous piecewise affine homeomorphism of $[0,1]$ .

(iv) By the previous property (iii), $\tilde G\in \tilde {\mathcal {F}}(\mathbb {R})$ . Lemma 2.6(ii) and (i) furthermore imply that $\tilde G\in \tilde {\mathcal {F}}_{\lambda }(\mathbb {R})$ and $g\in C_{\tilde \lambda }(\mathbb {S}^1)$ .

(v) This is because the slopes of piecewise affine outer homeomorphism h can change only at the points from $F(T(\tilde F))$ .

(vi) For each interval $(u,v)$ , where

1. (1) either $u=0$ and v is the least value $F(x)>0$ at a turning point x of $\tilde F$ ,

2. (2) or $u,v$ are two consecutive values at turning points of $\tilde F$ ,

3. (3) or u is the biggest value $F(x)<1$ at a turning point x of $\tilde F$ and $v=1$ ,

$F^{-1}((u,v))$ can be expressed as a finite union

$$ \begin{align*}F^{-1}((u,v))=\bigcup_{j}(a_j,b_j)\quad\text{where }F((a_j,b_j))=(u,v),\text{ for each }j. \end{align*} $$

It follows from our definition of $\tilde {\mathcal {F}}_0(\mathbb {R})$ in (7) that $a_j,b_j\in \mathbb {Q}_{\pi }$ for all j, so

(9) $$ \begin{align} \lambda(F^{-1}((u,v)))=\sum_j(b_j-a_j)\in\mathbb{Q}. \end{align} $$

Fix a turning point $w\in T_{[0,1]}(\tilde F)$ for which $F(w)>0$ . One can set

$$ \begin{align*}0=u_1<v_1=u_2<v_2=\cdots=v_k=F(w), \end{align*} $$

where $u_i,v_i$ were described above in (vi)(1)–(3); then by (8) and (9),

(10) $$ \begin{align} G(w)=(h\circ F)(w)&=\lambda(F^{-1}(0,F(w)))=\sum_i\lambda(F^{-1}((u_i,v_i)))\\ &=\sum_i\sum_j(b_j-a_j)\in\mathbb{Q}\nonumber. \end{align} $$

By (v) and (7)(ii), $D(\tilde G)\subset D(\tilde F)\cap [0,1]= F^{-1}(S)\subset \mathbb {Q}_{\pi }$ . Since $G(x)\in \mathbb {Q}$ if and only if $\tilde G(x)\in \mathbb {Q}$ , from (10) and (7)(ii) we obtain $\tilde G(D(\tilde G))=\tilde G(D(\tilde F))\subset \mathbb {Q}$ , hence

$$ \begin{align*}\tilde G(D(\tilde G))\cap D(\tilde G)=\emptyset.\end{align*} $$

(vii) This property is a consequence of (iv) and the fact that $\tilde G$ is a lifting of g due to formula (1).

To prove (viii), by conditions (v) and (vi) we can assume that $x\notin D(\tilde {F})$ . Let $x\in (p,q)\cap \mathbb {Q}_{\pi }$ , where $p,q\in D(\tilde F)$ are adjacent. Then

$$ \begin{align*}\tilde G(x)=\tilde G(p)+\frac{\tilde G(q)-\tilde G(p)}{q-p}(x-p). \end{align*} $$

By (vi), each of the numbers $\tilde G(p)$ , $\tilde G(q)-\tilde G(p)$ , $q-p$ and $x-p$ is rational, so $\tilde G(x)\in ~\mathbb {Q}$ .

Using Lemma 3.2 we introduce the set $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ of circle maps from $C_{\tilde \lambda }(\mathbb {S}^1)$ with liftings in $\tilde {\mathcal {F}}_{\lambda ,0}(\mathbb {R})$ , where

(11) $$ \begin{align}\tilde{\mathcal{F}}_{\lambda,0}(\mathbb{R}):=\{\tilde G\colon~\tilde F\in \tilde{\mathcal{F}}_0(\mathbb{R})\text{ and }G=h\circ F\}. \end{align} $$

Recall that by our definition the set $\tilde {\mathcal {F}}(\mathbb {R})$ consists of liftings of onto circle maps (see Remark 2.4 and the text preceding it).

Proof. Fix $\varepsilon>0$ and a map $e\in C_{\tilde \lambda }(\mathbb {S}^1)$ with a lifting $\tilde E\in \tilde {\mathcal {F}}_{\lambda }(\mathbb {R})$ . By Lemma 3.1 and Remark 2.4 there is a map $\tilde F\in \tilde {\mathcal {F}}_0(\mathbb {R})$ such that its representative $F=\tilde F\vert [0,1)$ is onto and:

1. (i) $\rho (\tilde E,\tilde F)< {\varepsilon }/2$ ;

2. (ii) for $h\colon [0,1]\to [0,1]$ defined by $h(0)=0$ , $h(x)=\lambda (F^{-1}((0,x)))$ (as in (8)),

   $$ \begin{align*} \rho(h,\mathrm{id})< \frac{\varepsilon}{2}.\end{align*} $$

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 $\text {id}$ ; we refer the reader to [Reference Bobok4] where the analogous interval case had been treated in detail. We have proved in Lemma 3.2(iv) that $G=h\circ F$ is a representative of a map g from $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ . Moreover, using (i) and (ii) and the definition of $\tilde G$ in Lemma 3.2(iii) and (iv) showing that $\rho (\tilde F,\tilde G)=\rho (F,G)$ , we obtain

$$ \begin{align*} \rho(e,g)&\le \rho(\tilde E,\tilde G)\le \rho(\tilde E,\tilde F)+\rho(\tilde F,\tilde G) \\ &=\rho(\tilde E,\tilde F)+\rho(F,G)=\rho(\tilde E,\tilde F)+\rho(F,h\circ F)<\varepsilon. \end{align*} $$

Thus, for each $\varepsilon>0$ and $e\in C_{\tilde \lambda }(\mathbb {S}^1)$ we have found a map $g\in C_{\tilde \lambda ,0}(\mathbb {S}^1)$ such that $\rho (e,g)<\varepsilon $ .

Definition 3.4 We say that two maps $f,g\colon ~[a,b]\subset [0,1]\to \mathbb {R}$ are $\lambda $ -equivalent if for each Borel set $A\subset \mathbb {R}$ ,

$$ \begin{align*}\lambda(f^{-1}(A))=\lambda(g^{-1}(A)). \end{align*} $$

For $0\le a<b\le 1$ and $0\le c<d\le 1$ we denote by $^+h_{{[a,b];[c,d]}},^-h_{[a,b];[c,d]}$ the affine maps from $[a,b]$ onto $[c,d]$ fulfilling $^+h(a)=c$ , $^+h(b)=d$ (respectively, $^-h(a)=d$ , $^-h(b)=c$ ).

Lemma 3.5 For any map $f\in C_{\lambda }(I)$ , the maps

$$ \begin{align*}^{\pm}h_{[0,1];[c,d]}\circ f\circ\kern0.5pt ^+h_{[a,b];[0,1]}\end{align*} $$

are $\lambda $ -equivalent to the maps $^{\pm }h_{[a,b];[c,d]}$ .

Proof. For each Borel $A\subset [c,d]$ we have

$$ \begin{align*} &\lambda((^{\pm}h_{[0,1];[c,d]}\circ f\circ\kern1pt ^{+} h_{[a,b];[0,1]})^{-1}(A)) \\ &\quad =\lambda((^+h_{[a,b];[0,1})^{-1}\circ f^{-1}\circ (^{\pm}h_{[0,1];[c,d]})^{-1}(A))\\ &\quad =\frac{b-a}{d-c}\lambda(A)=\lambda((^{\pm}h_{[a,b];[c,d]})^{-1}(A)). \\[-42pt] \end{align*} $$

We will apply Lemma 3.5 for two special classes of elements from $C_{\lambda }(I)$ . The first consists of piecewise affine maps with $2n+1$ , $n\in \mathbb {N}$ , full laps: for points $0=x_0<x_1<\cdots <x_{2n}<x_{2n+1}=1$ , $\bar {x}=(x_0,x_1,\ldots ,x_{2n+1})$ and $i\in \{0,1,\ldots ,2n+1\}$ we define $\beta [2n+1,\bar {x}]\in C_{\lambda }(I)$ as

(12) $$ \begin{align} \beta[2n+1,\bar{x}](x_{i}) := \begin{cases} 0,\quad& i\text{ even},\\ 1,\quad& i\text{ odd}, \end{cases} \end{align} $$

and continuous, affine on each $[x_i,x_{i+1}]$ .

The second class that we define consists of maps $\Psi =\Psi [\varepsilon ,a',d,e,h']$ , where

(13) $$ \begin{align} 0<\frac{\varepsilon}{3}<a',\quad a'+\frac{2\varepsilon}{3}<d<e<h'-\frac{2\varepsilon}{3},\quad h'<1-\frac{\varepsilon}{3}, \end{align} $$

as illustrated by the right-hand part of Figure 4 and its caption.

Figure 4 For $\varepsilon \in (0,1/2)$ , $a=b-a=c-b=g-f=h-g=1-h=b'-a'=c'-b'=g'-f'=h'-g'= {\varepsilon }/{3}$ .

Proof. Consider the discontinuous map $\Phi $ defined by the left-hand part of Figure 4. The reader can easily verify that for each $y\in (0,1)\setminus \{\varepsilon ,1-\varepsilon \}$ ,

(14) $$ \begin{align} \sum_{x\in \Phi^{-1}(y)}\frac{1}{\vert \Phi'(x)\vert}=1. \end{align} $$

Now pick the points $a,b,b',c,c',f,f',g,g',h\in [0,1]$ as suggested in Figure 4. The map $\Psi $ from the right-hand part of Figure 4 is continuous. Using its description in the caption of Figure 4, let us show that $\Psi \in C_{\lambda }(I)$ . It is clear that for any $y\in (0,\varepsilon )\cup (1-\varepsilon ,1)$ the equality (14) estimated for $\Psi $ holds true again. For any $y\in (\varepsilon ,1-\varepsilon )$ we can write

(15) $$ \begin{align} \sum_{x\in \Psi^{-1}(y)}\frac{1}{\vert \Psi'(x)\vert}&=\frac{a'-a}{1-2\varepsilon}+\frac{d-c'}{1-2\varepsilon}+\frac{e-d}{1-2\varepsilon}\\ &\quad +\frac{f'-e}{1-2\varepsilon}+\frac{h-h'}{1-2\varepsilon}=\clubsuit\nonumber. \end{align} $$

Since $a'-a+d-c'=d-c$ and $f'-e+h-h'=f-e$ , we can rewrite (15) with the help of (14) as

$$ \begin{align*} &\clubsuit=\frac{d-c}{1-2\varepsilon}+\frac{e-d}{1-2\varepsilon}+\frac{f-e}{1-2\varepsilon}=1, \end{align*} $$

that is, $\Psi \in C_{\lambda }(I)$ by Lemma 2.7.

3.3 Partitions, special perturbations

In this section we will start with maps from $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ defined in the previous section and particular associated partitions of $\mathbb {S}^1$ 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 Figure 5 to visualize the discussed concepts better.

Figure 5 $J=\phi ([q_i,q_{i+1}])=L^J_1\cup L^J_2\cup M^J\cup R^J_2\cup R^J_1$ .

Given a piecewise affine circle map $g\in C_{\tilde \lambda ,0}(\mathbb {S}^1)$ , $\varepsilon>0$ and its affine partition $\mathcal {P}\supset D(g)$ for which

$$ \begin{align*} \mathcal{P}\subset \phi(\mathbb{Q}_{\pi})\text{ and }\lVert\mathcal{P}\rVert<\varepsilon \end{align*} $$

(where $\lVert \cdot \rVert $ denotes the maximum diameter of partition elements), we will construct a perturbation $\theta $ of g and a partition $\mathcal {Q}\subset \phi (\mathbb {Q}_{\pi })$ for $\theta $ for which $\mathcal {P}\prec \mathcal {Q}$ (that is, $\mathcal {Q}$ refines $\mathcal {P}$ ) and such that each $J\in \mathcal {Q}$ 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 and satisfies the following conditions.

• (C1) There is $I\in \mathcal {Q}$ (depending on J) such that $\theta (J)\supset I$ .

• (C2) Let $I\in \mathcal {Q}$ be such that $\theta (J)\cap (L^I_1\cup L^I_2)\neq \emptyset $ . Then

  1. (a) $L^I_1\cup L^I_2\subset \theta (J)$ ,

  2. (b) if $K\in \mathcal {Q}$ is the unique element such that $R_1^K\cap L_1^I\neq \emptyset $ then $R^K_1\cup R^K_2\subset \theta (J)$ .

• (C3) Let $I\in \mathcal {Q}$ be such that $\theta (J)\cap (R^I_1\cup R^I_2)\neq \emptyset $ . Then

  1. (a) $R^I_1\cup R^I_2\subset \theta (J)$ ,

  2. (b) if $K\in \mathcal {Q}$ is the unique element such that $L_1^K\cap R_1^I\neq \emptyset $ then $L^K_1\cup L^K_2\subset \theta (J)$ .

• (C4) $\theta (J)=\theta (L_1^J)=\theta (R_1^J)$ .

• (C5) $B_{4\eta }(\theta (M^J\cup L_2^J\cup R_2^J))\subset \theta (J)$ for sufficiently small $\eta>0$ .

By (11), the map g has its lifting $\tilde G$ from ${\mathcal {F}}_{\tilde \lambda ,0}(\mathbb {R})$ represented by $G=h\circ F\in {\mathcal {F}}([0,1))$ , where F and h were described immediately prior to Lemma 3.2. By (iii) and (iv) of Lemma 3.2, g is piecewise affine, that is, such that the map $\tilde G$ (respectively, G) is piecewise affine. Applying (8) and Lemma 3.2, we can consider a finite set P of points such that

$$ \begin{align*}D(\tilde F)\cap [0,1]\subset P:=\{0<p_1<p_2<\cdots<p_m<1\}\subset \mathbb{Q}_{\pi}\end{align*} $$

for which $\tilde G\vert [p_i,p_{i+1}]$ is affine for each i (set $p_{m+1}=p_1+1$ ), and for

$$ \begin{align*} \mathcal{P}:=\phi(P)=\{\phi(p_1),\ldots,\phi(p_m)\} \subset\phi(\mathbb{Q}_{\pi})\subset \mathbb{S}^1, \end{align*} $$

1. (a) $g(\mathcal {P})\subset \phi (\mathbb {Q})$ hence $g(\mathcal {P})\cap \mathcal {P}=\emptyset $ ,

2. (b) $\lVert \mathcal {P}\rVert :=\max _{1\le i\le m}\vert \phi (p_{i+1})-\phi (p_i)\vert <\varepsilon $ .

We will call the set P (respectively, $\mathcal {P}$ ) a partition for $\tilde G$ (respectively, g). Redefining $\tilde G$ on each $[p_i,p_{i+1}]$ by (the numbers $n(i)\in \mathbb {N}$ and vector $\bar {x}(i)$ will be specified later)

(16) $$ \begin{align}\tilde \Sigma_i:=\kern0.5pt ^{s(i)}h_{[0,1];[\tilde G(p_i),\tilde G(p_{i+1})]}\circ \beta[2n(i)+1,\bar{x}(i)]\circ\kern0.5pt ^{+}h_{[p_i,p_{i+1}];[0,1]},\end{align} $$

where the $\beta $ s were introduced in (12) and $s(i)\in \{+,-\}$ are chosen to satisfy $\tilde \Sigma _i(p_i)=\tilde G(p_i)$ , yields a map $\tilde \Sigma \colon ~[0,1]\to \mathbb {R}$ given by $\tilde \Sigma (x)=\tilde \Sigma _i(x)$ , $x\in [\tilde {p}_i,\tilde {p}_{i+1}]$ . Notice that still $\tilde \Sigma (1)-\tilde \Sigma (0)=\mathrm {deg}(g)$ , so abusing the notation we will again denote by $\tilde \Sigma $ its extension from $[0,1]$ to the whole real line keeping the rule $\tilde \Sigma (x+1)=\tilde \Sigma (x)+\mathrm {deg}(g)$ . In fact the map $\tilde \Sigma $ is a lifting of some map $\sigma \colon ~\mathbb {S}^1\to \mathbb {S}^1$ . Because by Lemma 3.5 each map $\tilde G\vert [p_i,p_{i+1}]$ has been replaced by a $\lambda $ -equivalent map $\tilde \Sigma _i$ , it follows that the map $\Sigma \in {\mathcal {F}}([0,1))$ representing $\sigma $ satisfies the conditions of Lemma 2.7, hence by Lemma 2.6(i) we have that $\sigma \in C_{\tilde \lambda }(\mathbb {S}^1)$ . For the map $\tilde \Sigma \vert [0,1]$ we will consider a new partition

$$ \begin{align*}Q:=\bigcup_{i=1}^m\bigcup_{j=0}^{2n(i)+1} h^{-1}_{[p_i,p_{i+1}];[0,1]}(x_j(i))=: \{0=q_1<q_2<\cdots<q_{m'}=1\} \end{align*} $$

for some $m'\in \mathbb {N}$ , where the vectors $\bar {x}(i)=(x_0(i),x_1(i),\ldots ,x_{2n(i)+1}(i))$ will be chosen to satisfy $Q\subset \mathbb {Q}_{\pi }$ . Thus, the set Q contains P and also all new turning points of $\tilde \Sigma $ in $(0,1)$ being in $Q\setminus P$ . From our specific choice of $\beta $ s in (16) and Lemma 3.2(viii) we obtain

$$ \begin{align*}Q\subset\mathbb{Q}_{\pi}\quad\text{and}\quad\tilde \Sigma(Q)\subset \mathbb{Q};\end{align*} $$

denoting $\mathcal {Q}=\phi (Q)$ , we analogously obtain for $\sigma $ and $\mathcal {Q}$ ,

$$ \begin{align*}\mathcal{Q}\subset\phi(\mathbb{Q}_{\pi})\quad\text{and}\quad\sigma(\mathcal{Q})\subset \phi(\mathbb{Q}), \end{align*} $$

which implies that

$$ \begin{align*}\sigma(\mathcal{Q})\cap \mathcal{Q}=\emptyset. \end{align*} $$

At the same time the numbers $n(i)$ (recall that the number of full laps of $\beta $ is $2n(i)+1$ ) can be taken sufficiently large to satisfy, for each i and arcs $[\phi (q_i),\phi (q_{i+1})]=\phi ([q_i,q_{i+1}])$ ,

$$ \begin{align*} \#\{j\colon~\sigma([\phi(q_i),\phi(q_{i+1})]) \cap [\phi(q_j),\phi(q_{j+1})]\neq\emptyset\}\ge 3. \end{align*} $$

Up to now, using rescaled versions of $\beta $ s, we have perturbed the map $\tilde G$ (respectively, g) on the intervals $[p_i,p_{i+1}]$ (respectively, arcs $\phi ([p_i,p_{i+1}])$ ) to obtain the lifting $\tilde \Sigma $ of $\sigma \in C_{\tilde \lambda }(\mathbb {S}^1)$ .

In the last part of this proof we will proceed similarly: using rescaled versions of $\Psi $ s from Figure 4, we will perturb the map $\tilde \Sigma $ (respectively, $\sigma $ ) on the intervals $[q_i,q_{i+1}]$ (respectively, arcs $\phi ([q_i,q_{i+1}])$ ) to obtain the lifting $\tilde \Theta $ of $\theta \in C_{\tilde \lambda }(\mathbb {S}^1)$ .

Therefore, for each i define

(17) $$ \begin{align}\tilde \Theta_i:=\ ^{s(i)\kern-1pt}h_{[0,1];[\tilde \Sigma(q_i),\tilde\Sigma (q_{i+1})]}\circ \Psi[\varepsilon_i,a^{\prime}_i,d_i,e_i,h^{\prime}_i]\circ\hspace{2pt} ^{+\kern-0.5pt}h_{[q_i,q_{i+1}];[0,1]}, \end{align} $$

where $s(i)\in \{+,-\}$ is chosen to satisfy

(18) $$ \begin{align}\tilde \Theta_i(q_i)=\tilde \Sigma(q_i); \end{align} $$

using $\tilde \Theta _i$ , we can define the map $\tilde \Theta \colon ~[0,1]\to \mathbb {R}$ by $\tilde \Theta (x)=\tilde \Theta _i(x)$ , $x\in [q_i,q_{i+1}]$ . The reason why the degree-preserving extension of $\tilde \Theta $ to the real line is a lifting of a map $\theta \in C_{\tilde \lambda }(\mathbb {S}^1)$ is analogous to the reason above: the map $\theta $ is represented by the map $\Theta =\tilde \Theta \vert [0,1)(\mathrm {mod}~ 1)\in {\mathcal {F}}([0,1))$ that fulfils conditions of Lemma 2.7. Let us consider the map $\tilde \Theta $ (respectively, $\theta $ ) with respect to partition Q (respectively, $\mathcal {Q}$ ). For what follows we refer the reader to the right-hand graph in Figure 4. Taking $\varepsilon _i$ , $a^{\prime }_i$ , $d_i$ , $e_i$ and $h^{\prime }_i$ in (17) such that

(19) $$ \begin{align}h^{-1}_{[q_i,q_{i+1}];[0,1]}(\{a_i,a^{\prime}_i,b^{\prime}_i,c^{\prime}_i,d_i,e_i,f^{\prime}_i,g^{\prime}_i,h^{\prime}_i,h_i\})\subset\mathbb{Q}_{\pi},\end{align} $$

$\varepsilon _i$ 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,

$$ \begin{align*} \tilde \Sigma(Q)\cap [q_i,q_{i+1}]&=\tilde \Theta(Q)\cap [q_i,q_{i+1}]\\ &=\tilde\Theta(Q)\cap (h^{-1}_{[q_i,q_{i+1}];[0,1]}(d_i), h^{-1}_{[q_i,q_{i+1}];[0,1]}(e_i)). \end{align*} $$

Let us put for each i, $[q_i,q_{i+1}]$ and $H:=h_{[q_i,q_{i+1}];[0,1]}$ ,

$$ \begin{align*}&L^i_1=H^{-1}([0,c^{\prime}_i]),\quad L^i_2=H^{-1}([c^{\prime}_i,d^{\prime}_i]),\quad M^i=H^{-1} ([d^{\prime}_i,e^{\prime}_i]),\\ &R^i_2=H^{-1}([e^{\prime}_i,f^{\prime}_i]),\quad R^i_1=H^{-1}([f^{\prime}_i,1]); \end{align*} $$

then, using $\phi $ , we can transfer these sets to the arc

$$ \begin{align*}J=\phi([q_i,q_{i+1}])\end{align*} $$

by

$$ \begin{align*}&L^J_1=\phi(L^i_1),~L^J_2=\phi(L^i_2),\quad M^J= \phi(M^i),\\ &R^J_2=\phi(R^i_2),\quad R^J_1=\phi(R^i_1). \end{align*} $$

A sketch of this construction is given in Figure 5.

Since $\mathcal {Q}\subset \phi (\mathbb {Q}_{\pi })$ is a refinement of $\mathcal {P}$ , from (b) we obtain that $\lVert \mathcal {Q}\rVert <\varepsilon $ . By Lemma 3.2, $\theta (\mathcal {Q})\cap \mathcal {Q}=\emptyset $ and conditions (C1)–(C5) for the map $\theta \in C_{\tilde \lambda }(\mathbb {S}^1)$ with respect to

$$ \begin{align*}\mathcal{Q}=\mathcal{Q}_{\varepsilon,\theta}:=\phi(Q)\end{align*} $$

easily follow.

3.4 S-limit shadowing is generic in $C_{\tilde \lambda }({\mathbb {S}^1})$

For a given $\varepsilon>0$ , assume that $\theta $ , $\mathcal {Q}_{\varepsilon ,\theta }$ with $\lVert \mathcal {Q}_{\varepsilon ,\theta }\rVert <\varepsilon $ , and $\eta $ are provided in such a way that they satisfy conditions (C1)–(C5) and there is also $\delta =\delta (\theta )>0$ such that:

• (C6) $\delta <\eta $ and $2\delta <\operatorname {\mathrm {diam}} K$ for any $K\in \{L^J_1,L^J_2,M^J, R_1^J, R^J_2\}$ and any $J\in \mathcal {Q}_{\varepsilon ,\theta }$ .

In the previous section we described a special type of perturbation of a piecewise affine map $g\in C_{\tilde \lambda ,0}(\mathbb {S}^1)\subset C_{\tilde \lambda }(\mathbb {S}^1)$ leading to a circle map $\theta $ . The main property of $\theta $ was stated in Lemma 3.7. We now apply a similar approach to a dense sequence of piecewise affine maps from $C_{\tilde \lambda ,0}(\mathbb {S}^1)$ which is possible by invoking Lemma 3.3.

To that end, let $\Gamma :=\{g_m\}_{m\ge 1}\subset C_{\tilde \lambda ,0}(\mathbb {S}^1)$ be a dense sequence of maps in $C_{\tilde \lambda }(\mathbb {S}^1)$ such that:

• • each $g_m$ has an affine partition $\mathcal {P}_m\subset \phi (\mathbb {Q}_{\pi })$ satisfying $\lVert \mathcal {P}_m\rVert < {1}/{m}$ ;

• • for each $n\ge m$ , $g_n(\mathcal {P}_m)\cap \mathcal {P}_m=\emptyset $ .

Notice that the second property is guaranteed by Lemma 3.2(viii). Following the previous section we perturb $g_m$ to $\theta _m$ with $\varepsilon ={1}/{m}$ , corresponding partition $\mathcal {Q}_m:= \mathcal {Q}_{{1}/{m}, \theta _m}$ , $\eta _m$ , $\delta _m=\delta (\theta _m) < {1}/{m}$ and $U_m$ an open neighbourhood around $\theta _m$ in $C_{\tilde \lambda }(\mathbb {S}^1)$ such that:

• • $U_m\subset B_{\delta _{m}}(\theta _m)$ ;

• • the boundary of $U_m$ does not intersect $\Gamma $ .

Repeatedly using Lemmas 3.2(viii) and 3.3, we will proceed as follows to construct sequences $\{\mathcal {Q}_m\}_{m=1}^{\infty }$ , where each $\mathcal {Q}_m$ is a subset of $\phi (\mathbb {Q}_{\pi })$ , and $\{U_m\}_{m=1}^{\infty }$ .

1. (1) We perturb $g_1$ to $\theta _1$ to obtain $g_n(\mathcal {Q}_1)\cap \mathcal {Q}_1=\emptyset $ for each $n\ge 1$ .

2. (2) For $m>1$ , having already constructed the sets $\mathcal {Q}_i$ and $U_i$ , $i=1,\ldots ,m-1$ , in order to construct $\mathcal {Q}_m$ and $U_m$ we distinguish two possibilities:

   1. (a) either $g_m\notin \bigcup _{i=1}^{m-1}U_i$ , and then we construct $\mathcal {Q}_m$ and $U_m$ to fulfil

      $$ \begin{align*}g_n(\mathcal{Q}_m)\cap \mathcal{Q}_m=\emptyset\quad\text{for each }n\ge m,~\overline{U}_m\cap \bigcup_{i=1}^{m-1}\overline{U}_i=\emptyset;\end{align*} $$

   2. (b) or $g_m\in U_i$ , where $i\le m-1$ is the largest number with this property; denoting by $E(\mathcal {Q}_i)$ the set of $\phi $ -images of points defined in (19) for all $J\in \mathcal {Q}_i$ , a new partition $\mathcal {Q}_m\subset \phi (Q_{\pi })$ will fulfil

      $$ \begin{align*}E(\mathcal{Q}_i)\cup\mathcal{Q}_i\cup\mathcal{P}_m\prec \mathcal{Q}_m\quad\text{and}\quad\overline{U}_m\subset U_i\setminus\bigcup_{i<j\le m-1}\overline{U}_j.\end{align*} $$
      In particular, $\mathcal {Q}_m$ is an affine partition for $\theta _m$ which is a refinement of $\mathcal {Q}_i$ and $g_n(\mathcal {Q}_m)\cap \mathcal {Q}_m=\emptyset \text { for each }n\ge m$ .

   In addition, we require that the boundary of ${U}_m$ does not intersect $\Gamma $ ; this is possible since $\Gamma $ is countable.

Let us put $A_n=\bigcup _{m\ge n}U_m$ . Clearly, each $A_n$ is open and dense so the intersection

$$ \begin{align*}A=\bigcap_{n\ge 1}A_n=\bigcap_{n\ge 1}\bigcup_{m\ge n}U_m\end{align*} $$

is a dense $G_{\delta }$ set in $C_{\tilde \lambda }(\mathbb {S}^1)$ .

Proof Proof of Theorem 1.1

We will prove that each $\tau \in A$ has the s-limit shadowing property. By our definition of A, there is an increasing sequence $\{m(k)\}_{k=1}^{\infty }$ such that

$$ \begin{align*}U_{m(1)}\supset U_{m(2)}\supset\cdots,\quad\{\tau\}=\bigcap_{k=1}^{\infty}U_{m(k)}. \end{align*} $$

Let $\mathbf {x}=\{x_s\}_{s=0}^{\infty }$ be an asymptotic pseudo orbit. By Lemma 3.7 the partition $\mathcal {Q}_{m(i)}$ and $\delta =\delta _{m(i)}>0$ were chosen for $\alpha $ -shadowing with $\alpha =1/m(i)$ . Fix k so that

(20) $$ \begin{align}4/k<\delta.\end{align} $$

Let the partition $\mathcal {Q}_{m(j)}$ and $\gamma :=\delta _{m(j)}>0$ where $i<j$ be provided for $\beta $ -shadowing with $\beta =1/m(j)\le 1/k$ . Note that by condition (C6), $\gamma <1/k$ , hence together with (20) we obtain

(21) $$ \begin{align}\gamma+2/k<3/k<4/k<\delta.\end{align} $$

Assume for simplicity that $\mathbf {x}$ is a $\delta $ -pseudo orbit and it is a $\gamma $ -pseudo orbit for all $s\geq N-1$ for some N. Let $J_s\in \mathcal {Q}_{m(i)}$ and $Q_s\subset J_s$ be provided as in (1) and (2) of the proof of Lemma 3.7 for $\mathbf {x}$ , and by the same conditions, let $R_s\in \mathcal {Q}_{m(j)}$ , and let $W_s\subset R_s$ for $s\geq N-1$ be provided by the fact that $\tau \in U_{m(j)}\subset B_{\delta _{m(j)}}(\theta _{m(j)})$ . In particular,

(22) $$ \begin{align}\tau(Q_s)\supset Q_{s+1},~s\ge 0,\quad\tau(W_s)\supset W_{s+1},~s\ge N-1. \end{align} $$

First, if $W_N\subset \tau (Q_{N-1})$ then we can switch directly from the arc $Q_{N-1}$ used for $\alpha $ -tracing to the arc $W_N$ used for $\beta $ -tracing.

Now, assume that $W_N\setminus \tau (Q_{N-1})\neq \emptyset $ . Notice that $W_N\cap \operatorname {\mathrm {int}} (Q_N)=\emptyset $ , since otherwise

$$ \begin{align*}W_N\subset Q_N\subset \tau(Q_{N-1})\quad\text{because }\mathcal{Q}_{m(i)}\prec\mathcal{Q}_{m(j)},\end{align*} $$

which gives a contradiction. On the other hand, $x_{N-1}\in J_{N-1}$ and $d(x_N,\tau (x_{N-1}))<\gamma $ and $\operatorname {\mathrm {diam}} (R_N)<1/k$ . Also $W_N\subset R_N$ and $x_N\in R_N$ . Then if $\xi :=\gamma +1/k$ ,

$$ \begin{align*}W_N\subset R_N\subset B_{\xi} (\tau(J_{N-1}))\subset B_{\delta} (\tau(J_{N-1})),\end{align*} $$

where the last inclusion is a consequence of (21). Since $W_N$ is not included in the arc $\tau (Q_{N-1})$ and both diameters $\operatorname {\mathrm {diam}} L_2^{J_N}$ and $\operatorname {\mathrm {diam}} R_2^{J_N}$ are greater than $2\delta $ , the only possibility is that

$$ \begin{align*}W_N\subset L_2^{J_{N}}\cup M^{J_{N}}\cup R_2^{J_{N}}.\end{align*} $$

But by (C5) we have

$$ \begin{align*}B_{4\delta}(\theta_{m(i)}(W_N))\subset \theta_{m(i)}(J_N)=\theta_{m(i)}(Q_N),\end{align*} $$

and thus since $\tau \in U_{m(i)}$ ,

$$ \begin{align*}B_{3\delta}(\theta_{m(i)}(W_N))\subset \tau(Q_N)\end{align*} $$

and

(23) $$ \begin{align}B_{2\delta}(\tau(W_N))\subset \tau(Q_N).\end{align} $$

On the other hand, from (22) we obtain

(24) $$ \begin{align}W_{N+1}\subset \tau(W_N)\subset B_{2\delta}(\tau(W_N)). \end{align} $$

Gluing (23) and (24) together, we get

$$ \begin{align*}W_{N+1}\subset \tau(Q_N).\end{align*} $$

This allows us to switch from the arc $Q_{N}$ used for $\alpha $ -tracing to the arc $W_{N+1}$ used for $\beta $ -tracing.

Then, using the above construction inductively, we obtain that for every $\varepsilon>0$ , every $\tau \in \bigcap _{n\ge 1} A_n$ and every asymptotic pseudo orbit $\mathbf {x}=\{x_s\}_{s=0}^{\infty }$ which is a $\delta $ -pseudo orbit, we can find a sequence of closed arcs $I_s\subset \mathbb {S}^1$ with the following properties:

1. (1) $\tau (I_s)\supset I_{s+1} $ ;

2. (2) $\operatorname {\mathrm {diam}} (I_s\cup \{x_s\})<\varepsilon $ ;

3. (3) for every $\beta>0$ there is $N>0$ such that $\operatorname {\mathrm {diam}} (I_s\cup \{x_s\})<\beta $ for all $s>N$ .

Now it is enough to take any $z\in \bigcap _{s\ge 0} \tau ^{-s}(I_s)$ to $\varepsilon $ -trace and asymptotically trace $\mathbf {x}$ .