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Introduction
We use ℕ and ℂ to denote the set of natural numbers and a complex plane, respectively. The simbol i means an imaginary unit.
By continuum we mean a compact connected metric space. A topological space X is unicoherent provided that whenever A and B are closed, connected subsets of X such that X = A ∪ B, then A ∩ B is connected. A topological space is hereditarily unicoherent provided that each of its closed, connected subset is unicoherent. By a dendroid we mean an arcwise connected hereditarily unicoherent continuum. A dendrite is a locally connected continuum without subsets homeomorphic to a circle. We note that a dendrite is a locally connected dendroid. Also we notice that a circle is not a unicoherent continuum. So a dendroid and a dendrite do not contain subsets homeomorphic to the circle and they are one-dimensional continua.
Let X be a dendroid with a metric d. An arc is any set homeomorphic to the closed interval [0,1]. We notice that any two distinct points x,y ∈ X can be joined by a unique arc with endpoints x, y (see, e.g., [1], [2]). We denote by [x,y] an arc joining x and y and containing these points, (x,y) = [x,y] \ {x,y}, (x,y] = [x,y] \ {x} and [x,y) = [x,y] \ {y}.
The set X \ {p} consists of one or more connected set. Each such set is called a component of a point p.
Definition 1
A point p ∈ X is called to be
an end point of X if the set X \ {p} is connected;
a branch point of X if the set X \ {p} has at least three components.
If X is a dendrite then the set of branch points and the number of components of any point p ∈ X are at most countable (see [1, §51]). These statements are not true for dendroids.
Let f : X → X be a continuous map of a dendroid X. ω-limit set of a point x ∈ X is the set
\omega (x,f) = \{ z \in X:\, \exists \, {n_j} \in {\mathbb N},{n_j} \to \infty ,\mathop {\lim }\limits_{j \to \infty } {f^{{n_j}}}(x) = z\} .
Definition 2
A point x ∈ X is said to be
a periodic point of f if f n(x) = x for some n ∈ ℕ. When n = 1, we say that x is a fixed point of f ;
a recurrent point of f if x ∈ ω(x, f );
a non-wandering point of f if for any neighborhood U(x) of a point x there is a number n ∈ ℕ so that f n(U(x)) ∩ U(x) ≠ ∅.
Let Fix( f ), Per( f ), Rec( f ), Ω( f ) denote the set of fixed points of f , the set of periodic points of f , the set of recurrent points of f , the set of non-wandering points of f respectively. It is well known that
Fix(f) \subseteq Per(f) \subseteq Rec(f) \subseteq \bigcup\limits_{x \in X} \omega (x,f) \subseteq \Omega (f).
Definition 3
[1, §46] Let f : X → X be a continuous map of a dendroid X. A map f is said to be monotone if for any connected subset C ⊂ f (X), f−1(C) is connected.
Let f : X → X be a monotone map. Denote by f n the n-iterate of f ; that is, f0 = identity and f n = f ○ f n−1 if n ≥ 1. We note that f n is monotone for every n ∈ ℕ.
For monotone maps on dendrites the next statements are true.
Theorem 1
[3] Let f : D → D be a monotone map of a dendrite D. Then for any point x ∈ D,\omega (x,f) \subseteq \overline {Per(f)}
.
Theorem 2
[4] Let f : D → D be a monotone map of a dendrite D. Then\Omega (f) = \overline {Per(f)}
.
Theorem 3
[5] Let f : D → D be a monotone map of a dendrite D. Then for any point x ∈ D, ω(x, f ) is either a periodic orbit or a minimal Cantor set.
In the note we show that Theorems 1 – 3 do not true for monotone maps on dendroids. Theorem 4 shows that Theorems 1, 2 do not hold for such maps.
Theorem 4
There are a dendroid X1and a monotone map f1 : X1 → X1such that
\omega (x,{f_1}) \not\subseteq \overline {Per({f_1})} for some point x ∈ X1;
\Omega ({f_1}) \ne \overline {Per({f_1})}
.
The next Theorem shows that Theorem 3 does not true for monotone maps on dendroids.
Theorem 5
There are a dendroid X2and a monotone map f2 : X2 → X2such that for some point x ∈ X2, ω(x, f2) is a nondegenerate closed interval belonging to the set Fix( f2).
We note that there are continuous skew products of maps of an interval with a closed set of periodic points such that some their trajectories have a nondegenerate closed intervals as ω-limits sets (see, e.g., [6] – [11]).
Let K be a Cantor set on the closed interval [0,1], a point
p({1 \over 2},{1 \over 2} + {\bf{i}}) \in {\mathbb C}
. We set
{X_1} = \bigcup\limits_{e \in K} [p,e].
Note that X1 is a dendroid which is not a locally connected continuum in any point x ∈ X1 \ {p}.
II. Construction of the map f1 : X1 → X1.
We need the auxiliary map named binary adding machine.
Definition 4
Let Σ = {( j1, j2,...)} be the set of sequences, where ji ∈ {0,1}. We put a metric dΣ on Σ given by
{d_\Sigma }(({k_1},{k_2}, \ldots ),({j_1},{j_2}, \ldots )) = \sum\limits_{i = 1}^{ + \infty } {{\delta ({k_i},{j_i})} \over 2},
where δ (ki, ji) = 1, if ki ≠ ji and δ (ki, ji) = 0, if ki = ji. The addition in Σ is defined as follows:
({k_1},{k_2}, \ldots ) + ({j_1},{j_2}, \ldots ) = ({l_1},{l_2}, \ldots ),
where l1 = k1 + j1 (mod 2) and l2 = k2 + j2 + r1 (mod 2), with r1 = 0, if k1 + j1 < 2 and r1 = 1, if k1 + j1 = 2. We continue adding the sequences in this way.
The adding machine map σ : Σ → Σ is defined as follows: for any ( j1, j2, j3,...) ∈ Σ,
\sigma (({j_1},{j_2},{j_3}, \ldots )) = ({j_1},{j_2},{j_3}, \ldots ) + (1,0,0, \ldots ).
To define a map f1 : X1 → X1 we need two auxiliary maps.
1. Let h : K → Σ be any homeomorphism. We define a map τ : X1 → X1 as follows: τ : [p,e] → [p,h−1 ○ σ ○ h(e)] be a linear homeomorphism so that τ(p) = p , τ(e) = h−1 ○ σ ○ h(e).
According to lemma 6 we get the next properties of τ:
1.1. τ is a homeomorphism;
1.2. Per(τ) = Fix(τ) = {p};
1.3. x ∈ Rec(τ) \ Per(τ) for any point x ∈ X1 \ {p}.
2. Let e be any point from K and ϕ : [p,e] → [0,1] be any linear homeomorphism so that ϕ(p) = 1, ϕ(e) = 0.
We define a second auxiliary map g : X1 → X1 by the following way: for any point e ∈ K
g : [p,e] → [p,e] be a homeomorphism such that g(x) = ϕ−1 ○ x2 ○ ϕ(x) for any point x ∈ [p,e]. Then a map g has the next properties:
2.1. g is a homeomorphism;
2.2. Per(g) = Fix(g) = {p} ∪ K;
2.3. for any point e ∈ K and an arbitrary point x ∈ (p,e], ω(x,g) = {e}.
Now we set f1 = g ○ τ : X1 → X1. By properties of maps τ and g, we get the following statements:
f1 is a homeomorphism and so f1 is a monotone map;
Per( f1) = Fix( f1) = {p};
for any point x ∈ X1 \{p}, ω(x, f1) is a minimal Cantor set K, that is ω(x, f1) = K. Hence,
\omega (x,{f_1}) \not\subseteq \overline {Per({f_1})}
.
Ω( f1) = {p} ∪ K. So
\Omega ({f_1}) \ne \overline {Per({f_1})}
.
For any number n ∈ ℕ \ {sk}k≥1 there is a natural number k ≥ 0 such that sk < n < sk+1. It follows from (1) that for any k ≥ 0 every interval (sk;sk+1) contains 2k+2 − 3 natural numbers. For every k ≥ 0 and any number 1 ≤ j ≤ 2k+2 − 3 we define a vertical segmet Isk+j by the following way:
{I_{{s_k} + j}} = \left\{ {\matrix{{\left[ {{1 \over {{2^{{s_k} + j}}}};{1 \over {{2^{{s_k} + j}}}} + (1 - {j \over {{2^{k + 1}}}}){\bf{i}}} \right],} \hfill & {{\rm{if}}\, \, \, 1 \le j \le {2^{k + 1}} - 1;} \hfill \cr {\left[ {{1 \over {{2^{{s_k} + j}}}};{1 \over {{2^{{s_k} + j}}}} + {{j + 2 - {2^{k + 1}}} \over {{2^{k + 1}}}}{\bf{i}}} \right],} \hfill & {{\rm{if}}\, \, \, {2^{k + 1}} \le j \le {2^{k + 2}} - 3.} \hfill \cr } } \right.
It follows from (2) and (3), that for any number n ∈ ℕ ∪ {0} we defined a segment In. Now we set
{X_2} = [0,1] \cup [0,{\bf{i}}] \cup \bigcup\limits_{n = 0}^\infty {I_n}.
A continuum X2 is a dendroid, but it is not a dendrite because X2 is not a locally connected continuum in any point x ∈ (0,i]. You can see a dendroid homeomorphic to X2 on figure 1.
II. Construction of the map f2 : X2 → X2.
We define a monotone map f2 : X2 → X2 as follows:
f2(z) = z, if z ∈ [0,i];
f2(z) = z/2, if z ∈ [0,1];
f2 : Ij → Ij+1 be a linear homeomorphism such that f2(Ij) = Ij+1 for any number j ≥ 0.
III. Properties of f2.
f2 is a homeomorphism.
Per( f2) = Fix( f2) = [0,i].
We show that f2 is a continuous map.
It is evident that f2 is a continuous map in any point z ∈ X2 \ [0,i]. We’ll prove a continuity of f2 in any point z ∈ [0,i]. Let U(z) be an arbitrary neighborhood of a point z and let ɛ > 0 be a diameter of U(z). We take any number k ≥ 1 so that Isk ∩ U(z) ≠ ∅. Then by (3) and (iii) for any j ≥ sk and for any point x ∈ Ij|{\rm{Im}}\, {f_2}(x) - {\rm{Im}}\, x| \le {1 \over {{2^{k + 1}}}},
where Im* is the imaginary part of a complex number *. By (ii) and (iii),
|{\rm{Re}}\, {f_2}(x) - {\rm{Re}}\, x| = {1 \over {{2^{j + 1}}}} \le {1 \over {{2^{k + 1}}}},
where Re* is a the real part of a complex number *.
It follows from (4) and (5) that for any j ≥ sk and any point x ∈ Ij|{f_2}(x) - x| \le \sqrt {{1 \over {{2^{2(k + 1)}}}} + {1 \over {{2^{2(k + 1)}}}}} = {1 \over {{2^{2k + 1}}}}.
Let U1(z) ⊂ U(z) be a neighborhood of a point x with diameter ɛ/2k+1. Then by (6)f2(U1(z)) ⊆ U(z), that is f2 is a continuous map in a point z.
4. We show that ω(1 + i, f2) = [0,i].
Let z be any point from [0,i] and U(z) be an arbitrary neighborhood of a point z of diameter d. We take any natural number k1 so that
{1 \over {{2^{{k_1}}}}} < {d \over 2}.
Now we take any natural number K ≥ k1 such that IsK ∩U(z) ≠ ∅. According to the choice of k1 and (4) there is a natural number j ≥ 1 so that
{\rm{Im}}\, f_2^j\left( {{1 \over {{2^{{s_K}}}}} + {\bf{i}}} \right) \in \left( {{\rm{Im}}\, z - {d \over 2},{\rm{Im}}\, z + {d \over 2}} \right).
It follows from here that
f_2^{{s_K} + j}(1 + {\bf{i}}) \in U(z)
. So, z ∈ ω(1 + i, f2).
Thus, ω(1 + i, f2) = [0,i] = Fix( f2). Theorem 5 is proved.