Minimal Non-invertible Maps on the Pseudo-Circle

Abstract

In this article we show that R.H. Bing’s pseudo-circle admits a minimal non-invertible map. This resolves a conjecture raised by Bruin, Kolyada and Snoha in the negative. The main tool is a variant of the Denjoy–Rees technique, further developed by Béguin–Crovisier–Le Roux, combined with detailed study of the structure of the pseudo-circle. This is the first example of a planar 1-dimensional space that admits both minimal homeomorphisms and minimal noninvertible maps.

1 Introduction

In the late 1960s J. Auslander raised a question concerning the existence of minimal noninvertible maps. Since then examples of such maps have been given, but the question as for what spaces such maps exist remains particularly interesting for compact and connected spaces (continua). In the present paper we are going to give the first example of a 1-dimensional space in the plane that admits both minimal homeomorphisms and minimal noninvertible maps. In general, for compact spaces the existence of an invertible minimal map is independent of the existence of a noninvertible one, and there exist spaces that admit both types of maps, either one of them, or none [26]. In [1] Auslander and Yorke observed that the Cantor set admits minimal maps of both kinds. In Kolyada et al. [26] showed how to obtain minimal noninvertible maps on the 2-torus from a class of minimal homeomorphisms. An analogous result was obtained in [39] by Sotola and Trofimchuk for the Klein bottle. In [9] Clark, the first and the last mentioned authors showed the existence of both types of minimal maps for any compact finite-dimensional metric space that admits an aperiodic minimal flow. The first author showed in [10] the existence of minimal noninvertible maps for all compact manifolds of dimension at least 2, that admit minimal homeomorphisms. Auslander and Katznelson showed that the circle admits no minimal noninvertible maps, despite supporting minimal homeomorphisms [3]. In this context Bruin, Kolyada and Snoha asked the following question:

Question 1

[12] Is the circle the only infinite continuum that admits a minimal homeomorphism but no noninvertible minimal map?

In the same paper Bruin, Kolyada and Snoha conjectured that R.H. Bing’s pseudocircle might provide a counterexample (see also [38]).

Conjecture 1

[12] The pseudo-circle is another continuum that admits minimal homeomorphisms, but no noninvertible minimal maps.

Quite recently, in [16] a construction of a family of continua that admit minimal homeomorphisms, but no minimal noninvertible maps was given, resolving Question 1 (see also [11]). However, Conjecture 1 has remained open until now. The pseudocircle seems indeed a natural candidate for another counterexample to Question 1, as it shares several properties with the circle. Topologically it is a circle-like planar cofrontier, whose homeomorphism group contains periodic rotations [22], and it appears in various contexts in dynamical systems. Handel showed that it appears as both a minimal subset of a volume preserving, $C^\infty$-smooth planar diffeomorphism [19], and a minimal attractor of a $C^\infty$-smooth planar diffeomorphism. Herman [21] extended Handel’s construction to show that the pseudo-circle can bound an open disk in the complex plane, on which a smooth planar diffeomorphism is complex analytic and complex analytically conjugate to an irrational rotation, and on the complement of which the diffeomorphism is smoothly conjugate to such a rotation. This result was further improved by Chéritat who showed that the pseudo-circle can appear as the boundary of a Siegel disk [13]. In [8] the first and last author showed that it also appears as a Birkhoff-like attractor [8]. Other related constructions were given by Yorke and the second author in [23,24,25]. Most recently, a continuous decomposition of the 2-torus into pseudo-circles was given by Béguin, Crovisier, and Jäger [4], that is invariant under a homeomorphism of the torus semi-conjugate to an irrational circle rotation. In the present paper, relying on Handel’s construction, we adapt and modify the results of Béguin, Crovisier and Le Roux [5] to the very fine and complex structure of the pseudo-circle, to provide a counterexample to the aforementioned conjecture.

Theorem 1.1

The pseudo-circle admits minimal noninvertible maps.

The results of [5], that are discussed in more detail in Sect. 3, are a far-reaching generalization of the Denjoy–Rees technique [15, 34] that allowed Rees to construct the first examples of minimal torus homeomorphisms with positive entropy. These results are crucial to our construction. However, it should be noted that our proof is not a mere application of very general results of [5] for manifolds to a special case of a particular 1-dimensional pathological fractal. Quite a lot of care and effort is needed to make sure that the construction will not change the underlying space, that is the pseudo-circle, into a different one-dimensional space. The general approach that we take is as follows. First, blow up a forward orbit of Handel’s minimal homeomorphism h contained in the pseudo-circle P, to a null sequence of pseudo-arcs $\{P_n\}_{n=0}^\infty$, with $h(P_n)=P_{n+1}$ for each $n\in {\mathbb {N}}$ to obtain a “new” pseudo-circle Q (Thoerem 3.1). Second, define an equivalence relation ’$\equiv$’ between points in Q that identifies any two points in each pseudo-arc in the sequence. This relation is monotone, and closed since the sequence of pseudo-arcs in null. Now consider the quotient space $P'=Q_{/_\equiv }$. This new space $P'$ is a pseudo-circle (Proposition 4.1). The homeomorphism h induces a minimal map on $P'$, that is not invertible on $h^{-1}(P_0)_{/_\equiv }=h^{-1}(P_0)$ (Theorem 4.2), and the goal of the construction is achieved. Therefore the success of the construction relies on the ability to guarantee that Q is indeed the pseudo-circle. To this end, one could think of a straightforward application of the following proposition from [5].

Proposition 1.2

(Béguin et al. [5]) Let R be a homeomorphism on a compact manifold M, and x a point of M which is not periodic under R. Consider a compact subset D of M which can be written as the intersection of a strictly decreasing sequence $\{D_n:n\in {\mathbb {N}}\}$ of tamely embedded topological closed balls. Then there exist a homeomorphism $f: M\rightarrow M$ and a continuous onto map $\phi : M \rightarrow M$ such that $\phi \circ f = R \circ \phi$, and such that

• $\phi ^{-1}(x)=D$;

• $\phi ^{-1}(y)$ is a single point if y does not belong to the R-orbit of x.

However, this is not sufficient to obtain our result. The problem, again, is that ’inserting’ the pseudo-arcs ad hoc does not guarantee that the resulting space will not change, as there is even no guarantee that such an insertion would allow these pseudo-arcs to have empty interior in the resulting space, as is required for the pseudo-circle. Moreover, even if one can ensure that the special nontrivial fibres indeed have empty interior in the new space, one still has to ensure that the insertion is such that the resulting space remains circle-like (i.e. it can be covered by circular chains of arbitrarily small mesh; see Section. 2 for precise definitions). Consequently, we were lead to an adjustment of the method in [5] (see the comments preceding Lemma 3.4 for more details) and combine it with a careful analysis of the hereditary indecomposable structure of the pseudo-circle. To that end, the notion of an internal composant introduced by Krasinkiewicz in [27] came in handy (see Sect. 4). Nevertheless, our construction made us appreciate the Denjoy–Rees technique, as presented in [5], even more and there are clearly other novel applications of it to come.

2 Preliminaries and Some Previous Constructions

In this section we define the notation and terms we use as well as recall some results from the literature that are used in the arguments.

2.1 Basic Notation in Topology and Dynamics

For any planar set E, we denote by $\text {Int}(E)$ the interior of the set E. We work with the closed annulus $\mathbb {A}=S^1\times [0,1]$, and its lift to $\widetilde{\mathbb {A}}=\mathbb {R}\times [0,1]$, with $\pi$ denoting the lift function from $\widetilde{\mathbb {A}}$ to $\mathbb {A}$.

Given a family ${\mathcal {E}}$ of connected subsets of $\mathbb {A}$, define

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}({\mathcal {E}})=\max \{ \text {diam} (X) \big | X\in {\mathcal {E}}\}. \end{aligned}$$

(2.1)

For any family ${\mathcal {E}}$ of subsets of $\mathbb {A}$, denote by ${\mathfrak {s}}({\mathcal {E}})$ the union of all the elements of ${\mathcal {E}}$, called the realization of ${\mathcal {E}}$. A non-degenerate continuum is a compact, connected metric space, which contains at least two elements. We call a continuum planar if it can be embedded into the plane. A continuum X is called indecomposable if X is not the union of two proper subcontinua. A continuum is hereditarily indecomposable if every subcontinuum of it is also indecomposable. For any $x\in X$, the composant of x, denoted C(x), is the union of all proper subcontinua in X containing x. Moreover, for any indecomposable continuum the following properties hold:

• If x, y are points in X, then either $C(x)=C(y)$ or $C(x)\cap C(y) = \emptyset$ ([33], Theorem 11.17),

• There exists uncountably many composants ([33], Theorem 11.15),

• Each composant is a dense first-category connected set in X.

Note that the first two properties do not hold for decomposable continua. Such continua have either one or three composants, and they are not pairwise disjoint ([33], Theorem 11.13). As for indecomposable continua, let us consider some examples well known from the theory of dynamical systems. The first class of such examples involves solenoids, which are defined as inverse limits of circles, with finite-to-one covering maps as bonding maps. Equivalently, solenoids can be defined as suspensions (mapping tori) of odometers. Smale used solenoids to provide one of the first examples of hyperbolic attractors [37]. Any solenoid ${\mathcal {S}}$ is an indecomposable continuum and composants of ${\mathcal {S}}$ coincide with arc-components of ${\mathcal {S}}$; i.e. the arc-component of any point is equal to the composant of that point. This follows directly from the fact that each arc-component is dense, and there are uncountably many such components, whose union is the entire solenoid. The same is true about the composants of derived from Anosov diffeomorphism attractors, Plykin attractors, and Denjoy minimal sets, with all three classes of continua (unlike the solenoids [7]) being embedable in surfaces (see e.g. [20], ([36], Appendix) and [31] for definitions). However, there are indecomposable continua that do not share this property, and in particular the pseudo-circle that we shall consider in the present paper contains no arcs at all(!), since it is hereditarily indecomposable. In particular, all arc-components of the pseudo-circle are degenerate, but its composants are unions of pseudo-arcs; see Remark 2.9.

Definition 2.1

Let X be a planar continuum. A composant C of X is called internal if every continuum L intersecting both C and the complement of X intersects all composants of X.

We will need the following theorem proved by Krasinkiewicz (see Main Theorems of [27, 28]).

Lemma 2.2

For any indecomposable planar continuum X, the union of all the internal composants is a $G_\delta$ subset of X. In particular, every indecomposable planar continuum contains an internal composant.

Let $f:X\rightarrow X, g:Y\rightarrow Y$ be two homeomorphisms on compact metric spaces X and Y, respectively. Let $\psi :X \rightarrow Y$ be a continuous surjective map satisfying $\psi \circ f = g \circ \psi$. In this case, we say $\psi$ is a semi-conjugacy between (X, f) and (Y, g). The map $\psi : X\rightarrow Y$ is called monotone if for any $y\in Y$, $\psi ^{-1}(y)$ is connected. We say the semi-conjugacy $\psi$ is almost 1-1 if there exists a residual subset $Y_1\subset Y$, such that for each $y\in Y_1$, $\phi ^{-1}(y)$ is a singleton.

Given a monotone map $\psi : X\rightarrow Y$, the partition space $\{\psi ^{-1}(y) \big | y \in Y\}$ is known to be upper semi-continuous, i.e., each set $\psi ^{-1}(y)$ is closed in X, and for any open set U in X, the union of all $\psi ^{-1}(y)$ contained in U is open. We need the following well known properties for upper semi-continuous decompositions. (See for example Proposition 1 and Proposition 2 in chapter 1 of [14].)

Lemma 2.3

Let $\phi : X\rightarrow Y$ be such that, $\{\phi ^{-1}(y)\big | y\in Y\}$ forms an upper semi-continuous decomposition of X. Then the following hold:

• (1) For any open set U containing some pre-image $\phi ^{-1}(y_0)$, the set $W= \bigcup _{y\in \phi (U)}\phi ^{-1}(y)$ is open.

• (2) Suppose a sequence of closed sets $\phi ^{-1}(y_n) \subset X$ converges to $A\subset X$ in the Hausdorff topology. Then for any $x\in A$, $A\subset \phi ^{-1} \circ \phi (x)$.

A Moore decomposition of the closed annulus $\mathbb {A}$ is an upper semi-continuous decomposition ${\mathcal {M}}$ such that any $M\in {\mathcal {M}}$ is contained in a topological disk. The following is the classic Moore’s theorem.

Lemma 2.4

(Section 25, Theorem 1 in [14]) For any Moore decomposition ${\mathcal {M}}$ on $\mathbb {A}$ , the partition space is again $\mathbb {A}$ . Moreover, there exists a continuous monotone surjection $\phi :\mathbb {A} \rightarrow \mathbb {A}$, such that the preimage of each point is precisely one element $M\in {\mathcal {M}}$.

2.2 Circular Chains and Pseudo-Circles

Let $\mathbb {Z}_m$ denote the finite additive Abelian group $\mathbb {Z}_m=\{{\overline{0}},\ldots , \overline{m-1}\}$. For any two elements ${\overline{i}},{\overline{j}} \in \mathbb {Z}_m$, we use the convenient notation $\rho$ for their distance. More precisely, denote $\rho (\overline{i},\overline{j})=\min ( |i-j|, |m-|i-j||)$.

Definition 2.5

For $m\ge 3$, a circular chain is a finite family of open subset ${\mathcal {D}}=\{D_{{\overline{\ell }}}\}_{{\overline{\ell }} \in \mathbb {Z}_m}$, not necessarily connected, such that

$$\begin{aligned} D_{{\overline{\ell }}} \cap D_{{\overline{k}}} \ne \emptyset \text { if and only if } \rho ({\overline{k}}, {\overline{\ell }})\le 1. \end{aligned}$$

(2.2)

Each element $D_{{\overline{\ell }}}$ is called a link.

Definition 2.6

Let ${\mathcal {D}}=\{D_{{\overline{\ell }}} : {\overline{\ell }} \in {\mathbb {Z}}_m\}$ and ${\mathcal {C}}=\{C_{{\overline{\ell }}} : {\overline{\ell }} \in {\mathbb {Z}}_s\}$ be two circular chains. We say ${\mathcal {D}}$ is crooked inside ${\mathcal {C}}$ if there exists a map $\pi :{\mathbb {Z}}_m\rightarrow {\mathbb {Z}}_s$ with the following properties.

• (i) For any ${\overline{i}} \in \mathbb {Z}_m$, the closure of $D_{{\overline{i}}}$ is contained in $C_{\pi ({\overline{i}})}$.

• (ii) With properly chosen circular orders, let ${\overline{i}}< {\overline{j}}$ be two elements in $\mathbb {Z}_m$, such that $\pi ({\overline{i}}) \le \pi ({\overline{j}})$. Suppose for any ${\overline{i}}<{\overline{k}}< {\overline{j}}$, $\pi ({\overline{i}}) \le \pi ({\overline{k}}) \le \pi ({\overline{j}})$. Then there exist ${\overline{i}}<{\overline{u}}<{\overline{v}}<{\overline{j}}$, such that,

  $$\begin{aligned}&\rho (\pi ({\overline{i}}),\pi ({\overline{v}}))\le 1. \end{aligned}$$

  (2.3)
  $$\begin{aligned}&\rho (\pi ({\overline{j}}),\pi ({\overline{u}}))\le 1. \end{aligned}$$

  (2.4)

We call a continuum X circle-like if for any $\varepsilon >0$, X can be covered by a circular chain ${\mathcal {C}}$ with ${{\,\mathrm{\text {mesh}}\,}}({\mathcal {C}})<\varepsilon$.

Definition 2.7

(See [6, 17] for the uniqueness part.) The pseudo-circle is the unique circle-like plane separating continuum that can be covered by a decreasing family of circular chains $\{{\mathcal {C}}_{n}\}_{n \ge 1}$, such that ${\mathcal {C}}_{n+1}$ is crooked inside ${\mathcal {C}}_{n}$ for each $n\ge 0$, and ${{\,\mathrm{\text {mesh}}\,}}({\mathcal {C}}_n)\rightarrow 0$ as n tends to $\infty$.

Remark 2.8

Alternatively, the pseudo-circle can also be characterised as circle-like, hereditarily indecomposable plane separating continuum which can be embedded in any two dimensional manifold (see [7]).

Remark 2.9

A pseudo-arc is similarly defined replacing the objects “circular chain” and “circle-like continua” by “linear chain” and “arc-like continua”, respectively. For precise definitions, see for example the book [33]. Here we want to mention several important facts. First, the pseudo-arc is also unique up to homeomorphism (see [6]). Then it follows from both definitions that any non-degenerate proper subcontinuum of a pseudo-circle is a pseudo-arc. Another useful fact is that, any non-degenerate subcontinuum of a pseudo-arc is also a pseudo-arc (this is the main theorem of [32]).

We also need the following Theorem from [22].

Lemma 2.10

(Theorem 3 of [22]) Let f be a homeomorphism of the pseudo-circle P. Let C be a composant of P and suppose $f(C)=C$. Then f admits a fixed point in C.

2.3 The Denjoy–Rees Technique: Settings

Denote by $\mathbb {A}= S^1\times [0,1]$ for the closed annulus. Let $f :\mathbb {A}\rightarrow \mathbb {A}$ be a homeomorphism which preserves both the orientation and the boundaries. The following definitions are adopted from [5].

Definition 2.11

For an integer $p\ge 1$, a finite family ${\mathcal {E}}$ of closed disks, contained in $S^1\times (0,1)$, is called p-iterable if for any $X,Y\in {\mathcal {E}}$ and integers $-p\le k,s\le p$, either $f^k(X)=f^s(Y)$ or $f^k(X)\cap f^s(Y)=\emptyset$.

For any p-iterable family of closed disks ${\mathcal {E}}$ and any $0\le n\le p$, we denote

$$\begin{aligned} {\mathcal {E}}^n=\bigcup _{|k|\le n} f^k({\mathcal {E}}), \end{aligned}$$

(2.5)

where $f({\mathcal {E}})=\{f(X) \big | X\in {\mathcal {E}}\}$. For any $0\le n < p$, define an oriented graph $G({\mathcal {E}}^n)$, where the vertices are elements of ${\mathcal {E}}^n$, and there is an edge from X to Y if and only if $f(X)=Y$. For a map h such that $h(X)\subset X$ and $h(Y)\subset Y$, we say that it commutes with f along edge $X\rightarrow Y$ if $f\circ h(x)=h\circ f(x)$.

Definition 2.12

Let ${\mathcal {E}},{\mathcal {F}}$ be two finite families of closed disks, all contained in the open set $S^1\times (0,1)$. By ${\mathfrak {s}}({\mathcal {E}})$ we denote the union of the elements of ${\mathcal {E}}$. We say ${\mathcal {F}}$ refines ${\mathcal {E}}$ if the following conditions hold.

• (1) Every element of ${\mathcal {E}}$ contains at least one element of ${\mathcal {F}}$.

• (2) For any $X\in {\mathcal {E}}$, $Y\in {\mathcal {F}}$, either $X\cap Y=\emptyset$ or $Y\subset {{\,\mathrm{int}\,}}(X)$.

Definition 2.13

Let ${\mathcal {E}},{\mathcal {F}}$ be two families of closed disks. For an integer $p\ge 0$, we say ${\mathcal {F}}$ is compatible with ${\mathcal {E}}$ for p iterates, if the following conditions hold.

• (1) ${\mathcal {E}}$ is p-iterable.

• (2) ${\mathcal {F}}$ is $(p+1)$-iterable.

• (3) ${\mathfrak {s}}({\mathcal {F}}) \subset {\mathfrak {s}}({\mathcal {E}})$, and ${\mathcal {F}}^{p+1}$ refines ${\mathcal {E}}^p$.

• (4) For every k with $|k|\le p$,

  $$\begin{aligned} {\mathfrak {s}}({\mathcal {F}}^{p+1}) \cap f^k( {\mathfrak {s}}({\mathcal {E}}) ) = f^k( {\mathfrak {s}}({\mathcal {F}}) ). \end{aligned}$$

  (2.6)

Let $\{{\mathcal {E}}_{(n)}\}_{n\ge 0}$ denote a sequence of families of closed disks. The following list of hypotheses was formulated in Section 2 of [5].

$\mathbf {A_1}$.:

For every $n \ge 0$,

(1):

${\mathcal {E}}_{(n)}$ is $(n+1)$-iterable and the graph $G({\mathcal {E}}_{(n)}^n)$ has no cycle.

(2):

${\mathcal {E}}^{n+1}_{(n)}$ refines ${\mathcal {E}}_{(m)}^{m+1}$ for any $0\le m <n$;

(3):

${\mathcal {E}}_{(n)}$ is compatible with ${\mathcal {E}}_{(n-1)}$ for n iterates.

$\mathbf {A_2}$.:

The following holds.

$$\begin{aligned} \lim _{n\rightarrow \infty }{{\,\mathrm{\text {mesh}}\,}}({\mathcal {E}}_{(n)}^n) = 0. \end{aligned}$$

(2.7)

Next, we want to choose a sequence of homeomorphisms $\{h_n\}_{n\ge 1}$, such that the following hold for every $n\ge 1$.

$\mathbf {B_{1}}$.:

The closure $\overline{\{x : h_n(x)\ne x\}}$ is contained in the set ${\mathfrak {s}}({\mathcal {E}}_{(n-1)}^{n-1})$.

$\mathbf {B_{2}}$.:

$h_n$ and f commute along edges of the graph $G({\mathcal {E}}_{(n-1)}^{n-1})$.

$\mathbf {B_3}$.:

The following holds.

$$\begin{aligned} \lim _{n\rightarrow \infty } {{\,\mathrm{\text {mesh}}\,}}\big (h_1^{-1}\circ \cdots \circ h_{n-1}^{-1} ({\mathcal {E}}_{(n)}^{n+1} \backslash {\mathcal {E}}_{(n)}^{n-1}) \big )=0. \end{aligned}$$

(2.8)

Whenever we have such a sequence of homeomorphisms $h_n$, for any $n\ge 1$, we define the homeomorphisms $\psi _n, g_n$ as follows.

$$\begin{aligned} \psi _n&=h_n \circ \ldots \circ h_1. \end{aligned}$$

(2.9)

$$\begin{aligned} g_n&=\psi _n^{-1}\circ f \circ \psi _n. \end{aligned}$$

(2.10)

Finally we set $\psi _0={{\,\mathrm{id}\,}}$ and $g_0=f$.

2.4 The Denjoy–Rees Technique: Results

The following results proved in [5] form the starting point of our proof. They ensure proper convergence of the sequence of conjugated homeomorphisms, while the conjugacies converge to a continuous map, which provides a semi-conjugacy. Later we will have to adjust this approach, when defining maps $h_n$ to ensure sufficiently fast convergence. This will be possible due to different approach to sets $X_n$ in (2.11). For purposes of [5] sets $X_n$ were specified as the beginning, and then the sequence was refined sufficiently fast. Unfortunately, this approach is likely to lead to a situation as on Fig. 1, destroying chainablity or other properties. To deal with this problem, sets $X_n$ will be defined dynamically as the construction progresses (and so the continuum in the fiber is unknown to the very end of the construction).

Fig. 1

Fiber nowhere dense in ${\mathbb {T}}^2$ but large in “blown up” continuum

Lemma 2.14

(Proposition 3.1 in [5]) Assume that hypotheses $\mathbf {A_{1}}, \mathbf {A_2}, \mathbf {B_{1}}, \mathbf {B_2}, \mathbf {B_3}$ are satisfied. Then:

• (1) The sequence $\{\psi _n\}_{n\ge 0}$ converges uniformly to a continuous surjective map $\psi :\mathbb {A}\rightarrow \mathbb {A}$.

• (2) The sequences $\{g_n\}_{n\ge 1}$ and $\{g_n^{-1}\}_{n\ge 1}$ converge uniformly to a homeomorphism $g :\mathbb {A}\rightarrow \mathbb {A}$ and its inverse $g^{-1}$, respectively.

• (3) The homeomorphism f is semi-conjugate to g via the map $\psi$.

Lemma 2.15

(Proposition 3.4 in [5]) Assume that hypotheses $\mathbf {A_{1}}, \mathbf {A_2}, \mathbf {B_{1}}, \mathbf {B_2}, \mathbf {B_3}$ are satisfied. Let $K=\bigcap _{n \ge 0}{\mathfrak {s}}({\mathcal {E}}_{(n)})$.

• (1) Suppose $x\in \mathbb {A}$ and there exists $m\in {\mathbb {Z}}$ such that $x\in f^m(K)$. Let $\{X_n\}_{n\ge |m|} \subset {\mathcal {E}}_{(n)}^{|m|}$ denote the decreasing sequence of closed disks containing x. Then

  $$\begin{aligned} \psi ^{-1}(x)=\bigcap _{n\ge |m|}\psi _n^{-1}(X_n). \end{aligned}$$

  (2.11)

• (2) For every x which does not belong to the orbit of K, $\psi ^{-1}(x)$ is a singleton.

3 Minimal Non-invertible Map on the Pseudo-Circle

The main result of this section is the following Theorem.

Theorem 3.1

There exists a homeomorphism $f:{\mathbb {A}}\rightarrow {\mathbb {A}}$ with an invariant pseudo-circle $Q \subset \mathbb {A}$ such that (Q, f) is minimal and there exists a pseudo-arc $A\subset Q$ such that

$$\begin{aligned} \lim _{|n|\rightarrow \infty }\text {diam} \big ( f^n(A) \big )=0. \end{aligned}$$

(3.1)

The proof of Theorem 3.1 goes through the whole section. In fact we will prove more than the statement here. We will provide a semi-conjugacy between two annulus homeomorphisms. The conclusion of the theorem is based on the existence of such a semi-conjugacy.

Let us give an outline before getting into details. In Sect. 3.1, we recall Handel’s construction of an annulus homeomorphism f, which admits an invariant pseudo-circle $P\subset \mathbb {A}$. Then, we construct the semi-conjugacy in Sect. 3.2 between $(\mathbb {A},f)$ and $(\mathbb {A},g)$ for some other homeomorphism g on $\mathbb {A}$. This is the most delicate part of the construction. In Sect. 3.3, we show the pre-image of P under the semi-conjugacy is still a pseudo-circle, and, finally, the rest of the proof of Theorem 3.1 is given in Sect. 3.4.

Based on Theorem 3.1, the proof of Theorem 1.1 is given in Sect. 4.

3.1 Minimal Homeomorphism on the Pseudo-Circle

Here we define a homeomorphism f on the closed annulus $\mathbb {A}$ which preserves a minimal invariant pseudo-circle P. The construction follows directly from [19], so we refer to this paper for more details (and some pictures for the construction). Roughly speaking, the pseudo-circle in construction in [19] is obtained as a nested sequence of annuli. Annulus in the n-th step of the construction is identified with $S^1\times [0,1]$ and then circle $S^1$ is split into exactly $m_n$ sub-intervals of equal lengths, say, $I^{(0)}_1,\cdots ,I^{(0)}_{m_n}$. This way we define circular chain ${\mathcal {C}}_n$, taking consecutive link of the form $(I^{(0)}_k\cup I^{(0)}_{k+1}) \times (0,1)$, for $k=1,\cdots , m_n-1$, and putting as the last $(I^{(0)}_{m_n}\cup I^{(0)}_1) \times (0,1)$. This allows us to express the main result of [19] in the following form.

Lemma 3.2

Let $\mathbb {A}=S^1\times [0,1]$. There exists a sequence of circular chains $\{{\mathcal {C}}_n\}_{n\ge 0}$ with links whose closures are closed discs, and a homeomorphism $f: \mathbb {A}\rightarrow \mathbb {A}$, with the following properties.

• (1) The following conditions are satisfied.

  $$\begin{aligned} \overline{{\mathfrak {s}}({\mathcal {C}}_0)}&\subset S^1 \times (0,1). \end{aligned}$$

  (3.2)
  $$\begin{aligned} \overline{{\mathfrak {s}}({\mathcal {C}}_{n+1})}&\subset {\mathfrak {s}}( {\mathcal {C}}_n), \text { for any }n\ge 0. \end{aligned}$$

  (3.3)

• (2) There is an f-invariant minimal pseudo-circle P such that

  $$\begin{aligned} P = \bigcap _{n\ge 0} {\mathfrak {s}}({\mathcal {C}}_{n}) = \bigcap _{n\ge 0} \overline{{\mathfrak {s}}({\mathcal {C}}_n)}. \end{aligned}$$

  (3.4)

3.2 Construction of the Semi-Conjugacy

In this subsection, we construct the semi-conjugacy. The main tool from this subsection is Lemma 3.4.

Lemma 3.3

For the homeomorphism $f:\mathbb {A}\rightarrow \mathbb {A}$ constructed in Lemma 3.2, there exists a sequence of pseudo-arcs $A_0\supset A_1 \supset \cdots$ contained in P, such that,

$$\begin{aligned} \lim _{n\rightarrow \infty } \text {diam}\big ( A_n\big ) =0. \end{aligned}$$

(3.5)

Moreover, for all $n\ge 0$, each $A_{n+1}$ is contained in an internal composant of $A_n$. In particular, $\bigcap _{n\ge 0} A_n$ is a singleton, containing a unique point, called $p_*$.

Proof

By Lemma 2.2, there exists an internal composant in P. So, we can choose $A_0\subset P$ to be a pseudo-arc in P, with diameter smaller than $2^{-1}$, which is contained in an internal composant of P. Then, for the same reason, we can choose $A_1$ to be a pseudo-arc contained in some internal composant of $A_0$, of diameter less than $2^{-2}$. Inductively, we can choose a sequence of pseudo-arcs, namely $\{A_n\}_{n\ge 1}$, such that, for any $n\ge 0$, the diameter $\text {diam}(A_n)$ is smaller than $2^{-n}$, and each $A_{n+1}$ is contained in an internal composant of $A_n$. The proof is completed. $\square$

Now we will perform a construction inspired by [5]. For our proofs it is essential to have both conditions (3.10) and (3.11) below satisfied and have a pseudoarc in one of the fibres. To do it following a scheme from [5] we would start with a continuum $A\subset P$ that we want to have in a fibre and then we use a decreasing family of its neighborhoods $U_n$ such that $\cap _n U_n=A$. However, if one checks the details carefully, then it becames evident that in this approach always one of the conditions (3.10) or (3.11) will be broken and it is impossible to satisfy them simultanously. Therefore we need an adjustment in the construction. The idea here is that we do not fix the seqence $U_n$ apriori, but construct it inductively ($U_{n+1}$ can be defined after $h_1,\ldots ,h_n$ are known). In the process, we have to ensure that the sequence of connected open set $U_n$ is nested (i.e. $\overline{U_{n+1}}\subset U_n$) and continuum obtained at the end $A=\cap _n U_n$ is within the family of continua that we are interested in (in our case, the pseudo-arc). These are main issues where we have to be especially careful.

Lemma 3.4

Denote by f, P and $p_*$ the objects obtained in Lemmas 3.2 and 3.3. Then there exist homeomorphisms $h_n$ on ${\mathbb {A}}$ such that if we denote

$$\begin{aligned} \psi _n:&=h_n\circ \cdots \circ h_1. \end{aligned}$$

(3.6)

$$\begin{aligned} g_n&= \psi _n^{-1}\circ f\circ \psi _n. \end{aligned}$$

(3.7)

the the following limits exist (in the topology of uniform convergence)

$$\begin{aligned} g:&=\lim _{n\rightarrow \infty } g_n. \end{aligned}$$

(3.8)

$$\begin{aligned} \psi :&= \lim _{n\rightarrow \infty }\psi _n \end{aligned}$$

(3.9)

and dynamical system $({\mathbb {A}},f)$ is a factor of $({\mathbb {A}},g)$ by the factor map $\psi$. For every $x\in {\mathbb {A}}$ not in the orbit of $p_*$ (i.e. $f^n(x)\ne p_*$ for every $n\in {\mathbb {Z}}$) the set $\psi ^{-1}(x)$ is a singleton, $\psi ^{-1}(p_*)$ is a pseudoarc and additionally there is an increasing sequence $k_\ell$ such that:

$$\begin{aligned} h_{\ell +1}^{-1} \big ( \overline{ {\mathfrak {s}}({\mathcal {C}}_{k_{\ell +1}}) }\big )&\subset {\mathfrak {s}}\big ( {\mathcal {C}}_{k_{\ell }} \big ). \end{aligned}$$

(3.10)

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}\big ( (h_{\ell }\circ \cdots \circ h_1)^{-1} ( {\mathcal {C}}_{k_{\ell }} ) \big )&< 2^{-\ell }. \end{aligned}$$

(3.11)

for every $\ell \ge 1$

Proof

Let $p_*\in \cap _n A_n\subset P$ be the pseudo-arc provided by Lemma 3.3. Recall that in the construction of f in Lemma 3.2, we have chosen a sequence of circular chains $\{{\mathcal {C}}_n\}_{n\ge 0}$, such that $P=\bigcap _{n\ge 0} {\mathfrak {s}}({\mathcal {C}}_n)$. Then, define $V_n$ to be an element contained in ${\mathcal {C}}_n$ which contains $p_*$. We also require (refining sequence ${\mathcal {C}}_n$ when necessary), that the elements of $\{f^i(\overline{V_n})\}_{i=-(n+2)}^{n+2}$ are pairwise disjoint and the sequence $V_n$ is nested.

As $U_0$ we take a neighborhood of the pseudo-arc $A_0$ such that $U_0\subset {\mathfrak {s}}({\mathcal {C}}_0)$ and $\{f^i(\overline{U_0})\}_{i=-2}^{2}$ has pairwise disjoint elements. Further sets $U_n$ will be defined inductively.

Regarding Lemmas 2.14 and 2.15 , we define inductively the following objects.

• (1) The sequence of families of closed disks, denoted by $\{{\mathcal {E}}_{(n)}\}_{n\ge 0}$, satisfying conditions $\mathbf {A_1}$ and $\mathbf {A_2}$ in Sect. 2.3.

• (2) The sequence of homeomorphisms $\{h_n\}_{n\ge 1}$ on the annulus $\mathbb {A}$ satisfying conditions $\mathbf {B_1}, \mathbf {B_2}, \mathbf {B_3}$ in Sect. 2.3.

We start by setting ${\mathcal {E}}_{(0)}=\{\overline{U_0}\}$, and $k_0=0$. It is immediate from definition that ${\mathcal {E}}_{(0)}$ satisfies conditions of $\mathbf {A_1}$ (only item (a) of $\mathbf {A_1}$ needs to be checked for $n=0$). Note that

$$\begin{aligned} {\mathcal {E}}_{(0)}^1=\{f^{-1}(\overline{U_0}), \overline{U_0}, f(\overline{U_0})\}. \end{aligned}$$

(3.12)

Let $U\subset \overline{U}\subset U_0$ be another neighborhood of $A_0$, such that $\overline{U}$ is homeomorphic to a closed disc, and is contained in the union of links of a chain cover of $A_0$ by discs of diameter at most $2^{-1}$. In other words, we may treat U as an open set and as a chain cover. Take any integers $s_1>0$ such that $V_{s_1}\subset U$.

Then we define the homeomorphism $h_1:\mathbb {A}\rightarrow \mathbb {A}$, with the following properties.

$$\begin{aligned} h_1 (U)&= V_{s_1}. \end{aligned}$$

(3.13)

$$\begin{aligned} h_1(p_*)&= p_*. \end{aligned}$$

(3.14)

$$\begin{aligned} h_1 \text { restricted to }&\mathbb {A}\backslash U_0 \text { is the identity}. \end{aligned}$$

(3.15)

Note that the condition $\mathbf {B_1}$ for $n=1$ follows from (3.15). Condition $\mathbf {B_2}$ is empty for $n=1$. Denote $\gamma ={{\,\mathrm{diam}\,}}A_0/2$. Let $k_1>0$ be an integer large enough, so that

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}( h_1^{-1} ( {\mathcal {C}}_{k_{1}} ) ) < 2^{-1} \end{aligned}$$

and put $U_{k_1}=U$. Define now

$$\begin{aligned} {\mathcal {E}}_{(1)} : = \{\overline{V_{s_1}} \}. \end{aligned}$$

(3.16)

By definition ${\mathcal {E}}_{(1)}$ is $(s_1+1)$-iterable, and $s_1\ge 1$. It also follows that ${\mathcal {E}}_{(1)}^2$ refines ${\mathcal {E}}_{(0)}^1$, and ${\mathcal {E}}_{(1)}$ is compatible with ${\mathcal {E}}_{(0)}$ for 1 iterate. In other words, the three conditions in $\mathbf {A_1}$ are all satisfied for $n=1$.

Now we will proceed with main construction, which will be repeated inductively.

Take any $m>k_1+s_1$ sufficiently large so that, if $R\ni p_*$ is a link of ${\mathcal {C}}_m$ then the sets $\{f^i(\overline{R}) : i=-3s_1,\ldots ,3s_1\}$ are pairwise disjoint and $\overline{R}\subset V_{s_1}$. Let s be any integer such that $A_s\subset R$.

Since in the definition of $h_{1}$ we only require (3.13) and (3.14), we can freely modify it on points of $U_{k_1}\supset A_0$, that is we may assume that ${{\,\mathrm{diam}\,}}h_1^{-1}(A_s)$ is arbitrarily close to ${{\,\mathrm{diam}\,}}U_{k_1}$, in particular exceeds $\gamma$.

Note that $A_s\subset P \subset {\mathfrak {s}}({\mathcal {C}}_m)$. Let $m'>m$ be such that

$$\begin{aligned} f^i({\mathfrak {s}}({\mathcal {C}}_{m'} ))\subset {\mathfrak {s}}({\mathcal {C}}_{m}) \qquad \text { for }|i|\le 1, \end{aligned}$$

and

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}h_{1}^{-1}({\mathcal {C}}_{m'})<2^{-2}. \end{aligned}$$

We also assume that $h_{1}^{-1}({\mathcal {C}}_{m'-1})$ defines a chain cover of $h_{1}^{-1}(A_s)$, which is a refinement of U, treated as a chain. Recall that ${\mathcal {C}}_{m'}$ is crooked inside ${\mathcal {C}}_{m'-1}$ by definition. Let $p_*\in W\subset \overline{W}\subset U_{k_1}$ be a neighborhood of $h_1^{-1}(A_s)$ obtained as the union of elements of a $2^{-2}$-chain cover of the pseudoarc $h_1^{-1}(A_s)$ consisting of elements of $h_{1}^{-1}({\mathcal {C}}_{m'})$ . Since links of ${\mathcal {C}}_{m'}$ are connected, W is connected as well.

If we view $U_{k_1}$ and W as chains, then we may assume that W is 1/4-chain crooked inside $U_{k_1}$ (in particular, it intersects all links that $U_{k_1}$ is union of). Let $W'$ be an open neighborhood of $A_s$ such that $\overline{W'}$ is homeomorphic to a closed disc and

$$\begin{aligned} A_s\subset W'\subset \overline{W'}\subset h_1(W)\subset V_{s_1} \end{aligned}$$

and take any $s_2>m'+k_1+s_1$ such that $\overline{V_{s_2}} \subset W'$.

Recall that the elements of $\{f^i(\overline{V_{s_2}}): |i|\le 2\}$ are pairwise disjoint sets. Then we can define the homeomorphism $h_2$ satisfying the following properties.

$$\begin{aligned} h_2(W')&=V_{s_2}, \end{aligned}$$

(3.17)

$$\begin{aligned} h_2(p_*)&=p_*, \end{aligned}$$

(3.18)

$$\begin{aligned} h_2 \text { commutes with } f&\text { on } \bigcup _{|i|\le 1} f^{i}(\overline{h_1(W)}), \end{aligned}$$

(3.19)

$$\begin{aligned} h_2 \text { restricted to }&\mathbb {A}\backslash \bigcup _{i=-1}^1 f^i(h_1(W)) \text { is the identity}. \end{aligned}$$

(3.20)

For (3.19), note that the sets $f^{-1}( V_{s_1} ), V_{s_1}$, and $f(V_{s_1})$ are pairwise disjoint, so first we define the restriction of $h_2$ to $\overline{W'}$ and then extend it to $V_{s_1}$ so that $h_2|_{(V_{s_1}{\setminus } h_1(W))}$ is the identity. Next put $f^i\circ h_2\big |_{V_{s_1}}\circ f^{-i}$ for $|i|\le 1$, and finally put $h_{2}(x)=x$ for all other points $x\in \mathbb {A}$. This way a homeomorphism $h_2$ on $\mathbb {A}$ is defined, which satisfies (3.19) and (3.20).

Now take $k_2$ sufficiently large, so that

$$\begin{aligned} h_{2}^{-1} \big ( \overline{ {\mathfrak {s}}({\mathcal {C}}_{k_{2}}) }\big )&\subset {\mathfrak {s}}\big ( {\mathcal {C}}_{k_{1}} \big ), \end{aligned}$$

(3.21)

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}(\big ( h_{2}\circ h_1)^{-1} ( {\mathcal {C}}_{k_{2}} ) \big )&< 2^{-2}, \end{aligned}$$

(3.22)

and put $U_{k_2}=h_1^{-1}(W')$.

As long as (3.22) is fairly easy to satisfy, (3.21) is possible since

$$\begin{aligned} \{x: h_2(x)\ne x\}\subset \cup _{|i|\le 1} f^i(h_1(W))\subset \cup _{|i|\le 1} f^i({\mathfrak {s}}({\mathcal {C}}_{m'})) \subset {\mathfrak {s}}({\mathcal {C}}_m) \subset {\mathfrak {s}}({\mathcal {C}}_{k_1}), \end{aligned}$$

in particular $h_2^{-1}(P)\subset {\mathfrak {s}}({\mathcal {C}}_{k_1})$. Note that $U_{k_2}$ is connected, so we also may view it as a chain with links induced by W. To finish the construction, put

$$\begin{aligned} {\mathcal {E}}_{(2)} : = \{\overline{V_{s_2}}\}. \end{aligned}$$

(3.23)

Directly from the definition of $h_2$, conditions $\mathbf {B_1}$ and $\mathbf {B_2}$ are satisfied for $n=2$. Sets $f^{i}(V_{s_2})$ are pairwise disjoint for $|i|\le s_2$ and $s_2\ge 4$, so we obtain that ${\mathcal {E}}_{(2)}$ is at least 4-iterable. Since $V_{s_2}$ is a neighborhood of $p_*$, it is easy to satisfy the requirement that ${\mathcal {E}}_{(2)}^3$ refines ${\mathcal {E}}_{(1)}^2$, and clearly also ${\mathcal {E}}_{(2)}$ is compatible with ${\mathcal {E}}_{(1)}$ for 2 iterates. Summing up, the three conditions in $\mathbf {A_1}$ are easily satisfied for $n=2$.

Assume now that for $i=1,2,\ldots ,n$ we have already defined increasing sequences $s_i,k_i$, open sets $U_{k_i}$ and maps $h_i:\mathbb {A}\rightarrow \mathbb {A}$ such that $h_i(p_*)=p_*$ and:

• (1) $U_{k_{i}}\subset \overline{U_{k_{i}}}\subset U_{k_{i-1}}$, $h_1\circ \ldots \circ h_i(U_{k_i})=V_{s_i}$ and for some $s>s_n$ we have $A_s\subset V_{s_n}$ and ${{\,\mathrm{diam}\,}}(h_1\circ \ldots \circ h_n)^{-1}(A_s)>\gamma$.

• (2) ${\mathcal {E}}_{(n)}=\{\overline{V_{s_n}}\}$ and conditions $\mathbf {A_1}$, $\mathbf {B_1}$, $\mathbf {B_2}$ and $\mathbf {B_3}$ are satisfied for each $i\le n$.

• (3) Each $\overline{U_{k_i}}$ is homeomorphic to a closed disc, we can present each set $U_{k_i}$ as a union of elements of a $1/2^i$-chain and under this representation $U_{k_{i}}$ is crooked in $U_{k_{i-1}}.$

• (4) The following conditions are satisfied:

  $$\begin{aligned} h_{i-1}^{-1} \big ( \overline{ {\mathfrak {s}}({\mathcal {C}}_{k_{i}}) }\big )&\subset {\mathfrak {s}}\big ( {\mathcal {C}}_{k_{i-1}} \big ). \end{aligned}$$

  (3.24)
  $$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}\big ( (h_i\circ \cdots \circ h_1)^{-1} ( {\mathcal {C}}_{k_{i}} ) \big )&< 2^{-i}. \end{aligned}$$

  (3.25)

Take any $m>k_n+s_n$ sufficiently large so that, if $R\ni p_*$ is a link of ${\mathcal {C}}_m$ then sets $\{f^i(\overline{R}) : i=-3s_n,\ldots ,3s_n\}$ are pairwise disjoint and $\overline{R}\subset V_{s_n}$. Let r be any integer such that $A_r\subset R$. The only requirements on $h_n$ are that $h_n(p_*)=p_*$ and $h_n(h_{n-1}\circ \ldots \circ h_1(U_{k_n}))=V_{s_n}$. Furthermore $A_r\subset A_{s}\subset V_{s_n}$ and ${{\,\mathrm{diam}\,}}(h_n\circ \cdots \circ h_1)^{-1} (A_s)>\gamma$. Therefore, we may adjust definition of $h_n$ on the set $h_{n-1}\circ \ldots \circ h_{1}(U_{k_n})$ in such a way that also ${{\,\mathrm{diam}\,}}(h_n\circ \cdots \circ h_1)^{-1} (A_r)>\gamma$ and all other assumed properties hold (in particular, we have to adjust it also on some images of $U_{k_n}$ so that commutativity in ${\mathbf {B}}_2$ remains satisfied).

Note that $A_r\subset P \subset {\mathfrak {s}}({\mathcal {C}}_m)$. Let $m'>m$ be such that

$$\begin{aligned} f^i({\mathfrak {s}}({\mathcal {C}}_{m'} ))\subset {\mathfrak {s}}({\mathcal {C}}_{m}) \qquad \text { for }|i|\le n, \end{aligned}$$

and

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}\big ( (h_n\circ \cdots \circ h_1)^{-1} ( {\mathcal {C}}_{m'} ) \big ) < 2^{-n-1}. \end{aligned}$$

We also assume that $(h_n\circ \cdots \circ h_{1})^{-1}({\mathcal {C}}_{m'-1})$ defines a chain cover of $(h_n\circ \cdots \circ h_{1})^{-1}(A_r)$, which is a refinement of $U_{k_n}$, treated as a chain. Let $p_*\in W\subset \overline{W}\subset U_{k_n}$ be a neighborhood of $(h_n\circ \cdots \circ h_1)^{-1} (A_r)$ obtained as the union of elements of a $2^{-n-1}$-chain cover of the pseudoarc $(h_n\circ \cdots \circ h_1)^{-1} (A_r)$ consisting of elements of $(h_n\circ \cdots \circ h_1)^{-1} ({\mathcal {C}}_{m'})$ . Since links of ${\mathcal {C}}_{m'}$ are connected, W is connected as well.

If we view $U_{k_n}$ and W as chains, then we may assume that W is $2^{-n-1}$-chain crooked inside $U_{k_n}$ (in particular intersects all links that $U_{k_n}$ is union of). Let $W'$ be an open neighborhood of $A_r$ such that $\overline{W'}$ is homeomorphic to a closed disc and

$$\begin{aligned} A_r\subset W'\subset \overline{W'}\subset h_n\circ \ldots \circ h_1(W)\subset V_{s_n} \end{aligned}$$

and take any $s_{n+1}>m'+k_n+s_n$ such that $\overline{V_{s_{n+1}}} \subset W'$.

Recall that the elements of $\{f^i(\overline{V_{s_{n+1}}}): |i|\le n+1\}$ are pairwise disjoint sets. Then we can define the homeomorphism $h_{n+1}$ satisfying the following properties.

$$\begin{aligned} h_{n+1}(W')&=V_{s_n}, \end{aligned}$$

(3.26)

$$\begin{aligned} h_{n+1}(p_*)&=p_*, \end{aligned}$$

(3.27)

$$\begin{aligned} h_{n+1} \text { commutes with } f&\text { on } \bigcup _{|i|\le n} f^{i}(\overline{h_n(W)}), \end{aligned}$$

(3.28)

$$\begin{aligned} h_{n+1} \text { restricted to }&\mathbb {A}\backslash \bigcup _{i=-n}^n f^i(h_n(W)) \text { is the identity}. \end{aligned}$$

(3.29)

For (3.28), note that the sets $f^{i}( V_{s_n} )$, $|i|\le n$ are pairwise disjoint, so first we define the restriction of $h_{n+1}$ to $h_n(W)\subset V_{s_n}$, extend it by identity to $V_{s_n}$, put $f^i\circ h_{n+1}\big |_{V_{s_n}}\circ f^{-i}$ for $i=-n,\ldots ,n$, and finally put $h_{n+1}(x)=x$ for all other points $x\in \mathbb {A}$. In this way a homeomorphism $h_{n+1}$ on $\mathbb {A}$, which satisfies (3.28) is defined.

Now take $k_{n+1}$ sufficiently large, so that

$$\begin{aligned} h_{n+1}^{-1} \big ( \overline{ {\mathfrak {s}}({\mathcal {C}}_{k_{n+1}}) }\big )&\subset {\mathfrak {s}}\big ( {\mathcal {C}}_{k_{n}} \big ), \end{aligned}$$

(3.30)

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}(\big ( h_{n+1}\circ \ldots \circ h_1)^{-1} ( {\mathcal {C}}_{k_{n+1}} ) \big )&< 2^{-n-1} \end{aligned}$$

(3.31)

and put $U_{k_{n+1}}=(h_{n+1}\circ \ldots \circ h_1)^{-1}(W')$. Note that $U_{k_{n+1}}$ contains $(h_n\circ \cdots \circ h_1)^{-1} (A_r)$, in particular ${{\,\mathrm{diam}\,}}(U_{k_{n+1}})>\gamma$.

As before (3.31) is easy to satisfy and (3.30) is possible, because

$$\begin{aligned} \{x: h_{n+1}(x)\ne x\}\subset & {} \cup _{|i|\le n} f^i(h_n\circ \ldots \circ h_1(W))\\\subset & {} \cup _{|i|\le n} f^i({\mathfrak {s}}({\mathcal {C}}_{m'})) \subset {\mathfrak {s}}({\mathcal {C}}_m) \subset {\mathfrak {s}}({\mathcal {C}}_{k_n}), \end{aligned}$$

in particular $h_{n+1}^{-1}(P)\subset {\mathfrak {s}}({\mathcal {C}}_{k_n})$. Note that $U_{k_{n+1}}$ is connected, so we also may view it as a chain with links induced by W. Additionally, modifying $h_{n+1}$ accordingly on $U_{k_{n+1}}$, we may assume that there is ${\hat{s}}>0$ such that $A_{{\hat{s}}}\subset V_{s_{n+1}}$ such that ${{\,\mathrm{diam}\,}}(h_{n+1}\circ \cdots \circ h_1)^{-1} (A_{{\hat{s}}})>\gamma$.

To finish the construction, put

$$\begin{aligned} {\mathcal {E}}_{(n+1)} : = \{\overline{V_{s_{n+1}}}\}. \end{aligned}$$

(3.32)

Directly from the definition of $h_{n+1}$, conditions $\mathbf {B_1}$ and $\mathbf {B_2}$ are satisfied for $n+1$. Sets $f^{i}(V_{s_{n+1}})$ are pairwise disjoint for $|i|\le s_{n+1}$ and $s_{n+1}\ge 2^{n+1}$, we obtain that ${\mathcal {E}}_{(n+1)}$ is at least $2^{n+1}$-iterable. Since $V_{s_{n+1}}$ is a neighborhood of $p_*$, it is easy to satisfy the requirement that ${\mathcal {E}}_{(n+1)}^{n+2}$ refines ${\mathcal {E}}_{(n)}^{n+1}$, and clearly also ${\mathcal {E}}_{(n+1)}$ is compatible with ${\mathcal {E}}_{(n)}$ for $n+1$ iterates. Summing up, the three conditions in $\mathbf {A_1}$ are easily satisfied for $n+1$. This completes the induction.

Since family of sets $V_n$ intersects in a single point, condition $\mathbf {A_2}$ is satisfied, proving all the properties claimed for the construction.

Define for all $n\ge 1$,

$$\begin{aligned} \psi _n:&=h_n\circ \cdots \circ h_1. \end{aligned}$$

(3.33)

$$\begin{aligned} g_n&=\psi _n^{-1}\circ f\circ \psi _n. \end{aligned}$$

(3.34)

Then, we apply Lemma 2.14, to obtain a homeomorphism g and a semi-conjugacy $\psi$, as the limits

$$\begin{aligned} g:&=\lim _{n\rightarrow \infty } g_n. \end{aligned}$$

(3.35)

$$\begin{aligned} \psi :&= \lim _{n\rightarrow \infty }\psi _n. \end{aligned}$$

(3.36)

Observe that the set $A=\cap _nU_{k_n}$ is continuum, as intersection of decreasing family of continua, all of diameter at least $\gamma$. As we announced in the construction, all sets $U_{k_n}$ can be viewed as $2^{-n}$-chains, so A is chainable, and each $U_{k_{n+1}}$ is crooked inside $U_{k_n}$, so A is hereditarily indecomposable, therefore a pseudoarc (see [6]).

Now we apply Lemma 2.15, obtaining that

$$\begin{aligned} \psi ^{-1}(y) \text { is a singleton for any } y&\notin \overline{\{f^n(p_*)\}}_{n\in \mathbb {Z}}. \end{aligned}$$

(3.37)

$$\begin{aligned} \psi ^{-1}(p_*)&=\bigcap _{n\ge 0}\psi _n^{-1}(\overline{V_{s_n}})=\bigcap _{n\ge 0} U_{k_n}=A. \end{aligned}$$

(3.38)

The proof of Lemma 3.4 is completed. $\square$

3.3 Properties of “Blow Up” of P

In this subsection, we prove the following.

Theorem 3.5

Let $\psi$ denote the semi-conjugacy map $\mathbb {A}$ obtained by construction in Lemma 3.4, and denote $Q=\psi ^{-1}(P)$. Then Q is a pseudo-circle.

Before we can prove the above result, we again need some preparation.

Recall that by Lemma 3.4 all preimages $\psi ^{-1}(x)$ for $x\in \mathbb {A}$ are connected, so $\psi$ is a monotone map. It follows immediately that Q is a continuum.

Lemma 3.6

The pseudo-arc $A=\psi ^{-1}(p_*)\subset Q$ is nowhere dense in Q.

Proof

Suppose to the contrary that A contains interior in Q. Equivalently, there exists an open subset $W\subset S^1\times (0,1)$ such that $W\cap Q \ne \emptyset$ and $W \cap Q=W \cap A$. Now we consider some connected component U of $W\backslash Q$ such that $\overline{U}\cap Q\ne \emptyset$. Since Q is closed, U is open. We can choose an arc $\alpha :[0,1] \rightarrow W$ connecting some point in U to $\partial U\cap Q$. More precisely, we can define $\alpha : [0,1]\rightarrow W$, such that, $\alpha (0) \in A$ and $\alpha \big ((0,1] \big ) \subset U$. In particular, $\alpha \big ( (0,1] \big ) \cap Q= \emptyset$. Then $\psi \big |_{ \alpha ([0,1])}$ is a homeomorphism, and by the construction of $\psi$ (see Lemma 3.4), $\psi \circ \alpha (0)=p_*$, while $\psi \circ \alpha \big ( (0,1] \big ) \cap P = \emptyset$. However, $\psi \circ \alpha : [0,1] \rightarrow \psi (W)$ is an arc containing both $p_*$ and some point in the complement of P. Since $p_*$ is contained in an internal composant of P, $\psi \circ \alpha \big ([0,1] \big )$ intersects all composants of P. In particular it contains many points of P, which is a contradiction. The proof of the lemma is completed. $\square$

Lemma 3.7

There exists a dense $G_\delta$ subset $I \subset Q$, such that, for every point $x\in I$,

$$\begin{aligned} \psi ^{-1}\circ \psi (x) =\{x\}. \end{aligned}$$

(3.39)

Proof

Note that $f\big |_Q :Q\rightarrow Q$ is a homeomorphism, and by Lemma 3.6, A is nowhere dense in Q. Thus, $f^n(A)$ is nowhere dense for all $n\in \mathbb {Z}$. Consider the set $\bigcup _{n\in \mathbb {Z}}f^n(A)$, which is a countable union of nowhere dense closed sets. Denote the complement by $I= Q\backslash \bigcup _{n\in \mathbb {Z}}f^n(A)$. Then by the Baire category theorem, I as an intersection of a sequence of open and dense sets is a dense $G_\delta$ set. Moreover, it follows from (3.37) and (3.38) that, for any $x\in I$,

$$\begin{aligned} \psi ^{-1}\circ \psi (x) =\{x\}. \end{aligned}$$

(3.40)

The proof is completed. $\square$

Lemma 3.8

Q is an indecomposable continuum.

Proof

Suppose to the contrary that Q is decomposable. Then we can write $Q=K\bigcup L$, where K and L are proper subcontinua of Q. It follows that $\psi (K)$ and $\psi (L)$ are two subcontinua of P as well as $P=\psi (K)\bigcup \psi (L)$. Since P is indecomposible, it follows either $\psi (K)$ or $\psi (L)$ is equal to P. Suppose without loss of generality that $\psi (L)=P$. Now recall that $K \backslash L= Q\backslash L\ne \emptyset$ is open in Q. So by Lemma 3.7, there exists $x\in \big ( K \backslash L\big ) \cap I$, where $I=\big ( Q\backslash \bigcup _{n\in \mathbb {Z}}f^n(A)\big )$. In particular, $\psi ^{-1} \circ \psi (x)= \{x\}$. On the other hand, by assumption that $\psi (L)=P$, there always exists $y\in L$ such that $\psi (y)=x$. This provides a contradiction. Thus, Q is indecomposable. $\square$

Lemma 3.9

The continuum Q is hereditarily indecomposable.

Proof

By comparing definitions, let us first point out the fact that, the composants of Q are precisely the preimages of the composants of P under the semi-conjugacy $\psi$.

Suppose to the contrary that a non-degenerate subcontinuum $M\subset Q$ is decomposable. By Lemma 3.8M is proper, hence $Q\backslash M$ is open in Q, and by Lemma 3.7, $(Q \backslash M)\cap I\ne \emptyset$, where $I=Q\backslash \bigcup _{n\in \mathbb {Z}}f^n(A)$, restricted to which, the semi-conjugacy $\psi$ is 1 to 1. In particular $\psi (M)\ne P$, so $\psi (M)$ is a proper subcontinuum of P. It follows that, $\psi (M)$ is either a singleton or a pseudo-arc. It is not hard to see that $\psi (M)$ can not be a singleton, because otherwise, by the construction of $\psi$, $M\subset f^n(A)$ for some $n\in \mathbb {Z}$. Then, as a non-degenerate subcontinuum of the pseudo-arc $f^n(A)$, by Remark 2.9, M itself has to be a pseudo-arc, which is indecomposable.

Now we suppose $\psi (M)$ is a pseudo-arc. By assumption, there are proper subcontinua K and L of M, such that $K \cup L = M$. It follows that $\psi (K)$ and $\psi (L)$ are two subcontinua of $\psi (M)$, which in turn is a subcontinuum of P. They satisfy

$$\begin{aligned} \psi (K) \cup \psi (L)= \psi (M). \end{aligned}$$

(3.41)

Since $\psi (M)$ is hereditarily indecomposible, either $\psi (L)$ or $\psi (K)$ is equal to $\psi (M)$. Let us assume $\psi (L)=\psi (M)$ for definiteness. Then, for any $x\in K\backslash L$, there exists some $y\in L$, such that $\psi (x)=\psi (y)$. It follows that $x\notin I$, i.e., there exists some $n\in \mathbb {Z}$ such that $x\in f^n(A)$.

Lemma 3.8 tells us Q is indecomposable. Now by Lemma 2.10, $f\big |_Q$ is fixed point free, which implies that no two members of the sequence $\{f^n(A)\}_{n\in \mathbb {Z}}$ of pseudo-arcs belong to the same composant of Q. Thus, $K\backslash L$ is contained in $f^n(A)$ for some $n\in \mathbb {Z}$. Up to composing by $f^{-n}$ if necessary, we simply assume that $K\backslash L\subset A$. Since $K \backslash L=M\backslash L$ is open in M, there is some nonempty open set $W \subset S^1\times (0,1)$ such that,

$$\begin{aligned} W \cap (K \backslash L)= & {} W \cap M \ne \emptyset . \end{aligned}$$

(3.42)

$$\begin{aligned} W \cap L= & {} \emptyset . \end{aligned}$$

(3.43)

Since $\psi (K \backslash L)=p_*$, as in the proof of Lemma 3.6, we can find some arc $\beta : [0,1] \rightarrow W$ such that

$$\begin{aligned}&\beta (0) \in A. \end{aligned}$$

(3.44)

$$\begin{aligned}&\beta \big ((0,1] \big ) \cap M=\emptyset . \end{aligned}$$

(3.45)

Since $\psi (M)$ is a pseudo-arc in P containing the point $p_*$, for each $i\in \mathbb {Z}$, it follows $A_i\cap \psi (M)\ne \emptyset$. Since the sets $A_i$ are pseudo-arcs, it follows either $A_i\subset \psi (M)$ or $\psi (M)\subset A_i$. Since by Corollary 3.11, $\lim _{n\rightarrow \infty }\text {diam} (A_n)=0$, there is $k>0$ such that $A_k \subset \psi (M)$. Now, by Lemma 3.3, $A_{k+1} \subset A_k$ contains $p_*$, and $A_{k+1}$ is in an internal composant of $A_k$.

Thus, denote $\beta ' :=\psi \circ \beta \big ( [0,1] \big )$ which is an arc in $\mathbb {A}$ containing the point $p_*$. Since $\beta '$ is in an internal composant of $A_k$, it must intersect all the composants of $A_k$. Then there exists $y \in A_k \cap \beta '$, with $y \ne p_*$. But $y \in \psi (M)$, and there is some $x \in M$ such that $\psi (x)=y$. Then $x \in L$, because otherwise $\psi (x)=p_*$. But then $x\in W\cap L$ which is a contradiction to (3.43). So Q is hereditarily indecomposable, and the proof is completed. $\square$

Lemma 3.10

The continuum Q is circle-like and separates ${\mathbb {A}}$.

Proof

By the construction, denote by $\{{\mathcal {C}}_k\}_{k\ge 0}$ for the defining sequence of circular chains for P, and let $\{h_n\}_{n\ge 0}$ and $\{\psi _n\}_{n\ge 0}$ be the sequence of homeomorphisms constructed in Lemma 3.4. Since the sequence $\{{{\,\mathrm{\text {mesh}}\,}}\big ( {\mathcal {C}}_k \big )\}_{k\ge 0}$ converges to 0, we can choose an increasing integer subsequence $\{k_j\}_{j\ge 0}$, such that $k_0=0$ and for any $j\ge 0$ we have

$$\begin{aligned} h_{j+1}^{-1} \big ( \overline{ {\mathfrak {s}}({\mathcal {C}}_{k_{j+1}}) }\big )&\subset {\mathfrak {s}}\big ( {\mathcal {C}}_{k_{j}} \big ). \end{aligned}$$

(3.46)

$$\begin{aligned} {{\,\mathrm{\text {mesh}}\,}}\big ( \psi _{j+1}^{-1} ( {\mathcal {C}}_{k_{j+1}} ) \big )&< 2^{-j}. \end{aligned}$$

(3.47)

Now we denote ${\mathcal {D}}_j = {\mathcal {C}}_{k_j}$ for all $j\ge 0$. Clearly, $\{\psi _n^{-1} ({\mathcal {D}}_n)\}_{n\ge 0}$ is a sequence of circular chains, and (3.46) show inductively this sequence is in fact decreasing. By (3.47), it follows that the continuum $\bigcap _{n\ge 0} {\mathfrak {s}}({\mathcal {D}}_n)$ is circle-like. So we are left to show the following.

$$\begin{aligned} Q=\bigcap _{n=0}^{\infty } \psi _n^{-1}({\mathfrak {s}}( {\mathcal {D}}_n)). \end{aligned}$$

(3.48)

Choose any $x \in \bigcap _{n=0}^{\infty } \psi _n^{-1}({\mathfrak {s}}({\mathcal {D}}_n))$. By the choice, $\psi _n (x) \in {\mathfrak {s}}({\mathcal {D}}_n)$ for all $n\ge 0$. Noting (3.36) and (3.3), it follows that $\psi (x) \in {\mathfrak {s}}( {\mathcal {D}}_n)$ for each $n\ge 0$. Thus,

$$\begin{aligned} \psi (x) \in \bigcap _{n=0}^{\infty } {\mathfrak {s}}\big ( {\mathcal {D}}_n \big )=P. \end{aligned}$$

(3.49)

Then $x \in \psi ^{-1}\circ \psi (x) \subset \psi ^{-1}(P)=Q$ and so $\bigcap _{n=0}^{\infty } \psi ^{-1}({\mathfrak {s}}( {\mathcal {D}}_n)) \subseteq Q$.

Next, suppose $Q'=\bigcap _{n=0}^{\infty } \psi _n^{-1}({\mathfrak {s}}({\mathcal {D}}_n))$ is a proper subcontinuum of Q. By definition $\psi _n(p_*)=p_*$ for all $n\ge 1$ and therefore $\psi _n^{-1}(p_*)=p_*\in \psi _n^{-1}({\mathfrak {s}}( {\mathcal {D}}_n))$ for each $n\ge 0$, which yields that $p_*\in Q'$. Then $Q'$ as a proper subcontinuum of Q, must be contained in the composant $C(p_*)$ of Q that contains $p_*$. Then $\psi (Q')$ is a proper subcontinuum of P. So, $\psi (Q')$ is either a pseudo-arc or a singleton. In particular, $\psi (Q')$ does not separate $\mathbb {A}$.

On the other hand, we know $\overline{{\mathfrak {s}}({\mathcal {D}}_0)}$ separates $\mathbb {A}$. More precisely, the set $\mathbb {A} \backslash \overline{{\mathfrak {s}}( {\mathcal {D}}_0)}$ has two connected components, and we call them S and T, respectively. Since $\psi (Q')$ is not separating, there exists an arc $\beta :[0,1] \rightarrow \mathbb {A}$ such that $\beta (0)\in \psi (S)$ and $\beta (1)\in \psi (T)$, and $\beta ( [0,1])\bigcap \psi (Q')=\emptyset$. Let $L:=\psi ^{-1}(\beta ([0,1]))$. Then L is a continuum contained in $S^1\times (0,1)$. Moreover,

$$\begin{aligned} L \cap Q' = \emptyset . \end{aligned}$$

(3.50)

However, $L \bigcap \psi _n^{-1} \big ( \overline{ {\mathfrak {s}}( {\mathcal {D}}_n)} \big ) \ne \emptyset$ for any $n\ge 0$, since it intersects both S and T. Then for any $n\ge 0$, we choose $y_n \in L \bigcap \psi _n^{-1} \big ( \overline{ {\mathfrak {s}}( {\mathcal {D}}_n)} \big )$. By extracting a subsequence if necessary, we can assume the sequence $\{ y_n\}_{n\ge 0}$ converges to a point $y_*\in L$.

We can also deduce from (3.46) the following for all $n >k \ge 1$.

$$\begin{aligned} \psi _n^{-1}( \overline{ {\mathfrak {s}}( {\mathcal {D}}_n) }) =&\psi _{n-1}^{-1}h_n^{-1}\big ( \overline{{\mathfrak {s}}( {\mathcal {D}}_n)} \big )\nonumber \\ \subset&\psi _{n-1}^{-1}( {\mathfrak {s}}( {\mathcal {D}}_{n-1}))\subset \ldots \nonumber \\ \subset&\psi _k^{-1}( {\mathfrak {s}}( {\mathcal {D}}_{k})) . \end{aligned}$$

(3.51)

Therefore $y_n \in \psi _k^{-1} \big ( \overline{{\mathfrak {s}}\big ( {\mathcal {D}}_k\big )} \big )$ for any $n\ge k\ge 0$. It follows that $y\in \psi _k^{-1} \big ( \overline{{\mathfrak {s}}\big ( {\mathcal {D}}_k\big )} \big )$ for all $k\ge 0$. Thus $y\in L\bigcap Q'$, which is a contradiction to (3.50). This contradiction shows $Q'=Q$. We have completed the proof of the lemma. $\square$

Proof

(End of Proof of Proposition 3.5) It follows directly from Lemma 3.9, Lemma 3.10 and the characterization stated in Remark 2.8, that Q is a pseudo-circle. $\square$

Corollary 3.11

Let $f:\mathbb {A}\rightarrow \mathbb {A}$ be the homeomorphism given in Lemma 3.2 with semiconjugacy $\psi :(\mathbb {A},g)\rightarrow (\mathbb {A},f)$, mapping the pseudo-circle Q onto the pseudo-circle P provided by Lemma 3.4. Then, for the pseudo-arc $A=\psi ^{-1}(p_*)$ we have

$$\begin{aligned} \lim _{|n|\rightarrow \infty } \text {diam} \big (g^n(A) \big )=0. \end{aligned}$$

(3.52)

Proof

Note that the semi-conjugacy $\psi$ on $\mathbb {A}$ can be regarded as a continuous map from Q to P. Suppose to the contrary that for some integer subsequence $\{n_k\}_{k\ge 0}$ and some $\delta >0$, every pseudo-arc $f^{n_k}(A)$ has diameter at least $\delta$. Let us fix some point $x\in Q$ such that

$$\begin{aligned} \psi ^{-1} \circ \psi (x)=\{x\}. \end{aligned}$$

(3.53)

By minimality of f on P, we can extract a further subsequence if necessary, still called $\{n_k\}_{k\ge 0}$, such that $g^{n_k}(A)$ converges to some set K containing x. By definition $g^n(A)=\psi ^{-1}(f^n(p_*))$ and so by Lemma 2.3, $\psi ^{-1}\circ \psi (x)\supset K$. However, K has diameter at least $\delta$, which is a contradiction with (3.53). $\square$

3.4 Proof of Theorem 3.1

Proof of Theorem 3.1

We have already proved in Theorem 3.5 that Q is a pseudo-circle. By Lemma 3.6, any iterate $g^n(A)$ has empty interior in Q for any $n\in \mathbb {Z}$. Thus, the set $Q\backslash \bigcap _{n\in \mathbb {Z}}g^n(A)$ is a dense $G_\delta$ subset of Q. This means that the set of points $y \in Q$ such that $\psi ^{-1}\circ \psi (y)=\{y\}$ is a dense $G_\delta$-set. In particular, $\psi$ is an almost 1-to-1 semi-conjugacy between (Q, g) and (P, f). Therefore, it follows immediately that (Q, g) is minimal since (P, f) is minimal. (see for example [18]). Equation (3.1) was established in Corollary 3.11. The proof is completed. $\square$

4 Proof of the Main Theorem

In this section we finish the proof of our main result. It seems that it is known that a monotone image of the pseudo-circle is homeomorphic to it (see e.g. p. 91, [30]), but for completness we provide a short justification below.

Proposition 4.1

Suppose $\phi : \mathbb {A} \rightarrow \mathbb {A}$ is a monotone continuous surjective map. Assume there is a non-degenerate continuum $Y\subset \mathbb {A}$, such that $\phi ^{-1}(Y)=P$. Then Y is a pseudo-circle.

Proof

We need to show that Y is circle-like, plane separating and hereditarily indecomposable.

It is well known that a monotone image of a circle-like continuum is circle-like (see e.g. Lemma 9, [29]) Moreover Y is non-degenerate and $\phi ^{-1}(Y)= P$ is plane separating. By the continuity of $\phi$, it is immediate that Y is also plane separating. It remains to show that Y is hereditarily indecomposable. But this is also immediate, since if a subcontinuum K of Y were decomposable, then $\phi ^{-1}(K)$ would also be a decomposable continuum, resulting in a contradiction. This shows that Y is a pseudo-circle. $\square$

Theorem 1.1 follows directly from the following statement.

Theorem 4.2

There exists a continuous surjection $g':{\mathbb {A}}\rightarrow {\mathbb {A}}$ with an invariant pseudo-circle $P\subset {\mathbb {A}}$ such that $(g',P)$ is minimal but is not one-to-one.

Proof

Let $g:{\mathbb {A}}\rightarrow {\mathbb {A}}$ denote the homeomorphism provided by Proposition 3.1 minimal on pseudo-circle Q and let $A\subset Q$ be the associated pseudo-arc. Define the equivalence relation $\sim$ in ${\mathbb {A}}$, such that, $x\sim y$ if and only if either $x=y$, or $x,y\in g^n(A)$ for some $n\ge 0$. Since by Proposition 3.1,

$$\begin{aligned} \lim _{|n|\rightarrow \infty } \text {diam} \big ( g^n(A)\big ) =0, \end{aligned}$$

(4.1)

it follows that $\sim$ is a closed equivalence relation with connected equivalence classes. Moreover, it is not hard to see the family of equivalence classes form a upper semi-continuous family. Thus, by Moore’s theorem (see Lemma 2.4), there exists a semi-conjugacy $\pi :\mathbb {A}\rightarrow \mathbb {A}$, which collapses each equivalence class into one point in $\mathbb {A}$. Then it induces a map ${\hat{g}}:\mathbb {A}\rightarrow \mathbb {A}$, such that $\pi \circ g= {\hat{g}} \circ \pi$. Note that ${\hat{g}}$ is a continuous non-invertible surjection. Indeed, $\pi (g^{-1}(A))$ is nondegenerate but ${\hat{g}}(\pi (g^{-1}(A)))=\pi (A)$ is a point in $\pi (Q)$, so ${\hat{g}}$ is non-invertible on $\pi (g^{-1}(A))$. In addition, $\pi :({\mathbb {A}},g)\rightarrow ({\mathbb {A}},{\hat{g}})$ is a monotone surjective continuous map (a factor map). It follows that ${\hat{g}}$ must be minimal. Then, by Proposition 4.1, the proof is completed. $\square$