On Limit Sets of Monotone Maps on Dendroids

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.

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 ω(x,f)={z∈X: ∃ nj∈ℕ,nj→∞,limj→∞fnj(x)=z}.\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\} .

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)⊆Per(f)⊆Rec(f)⊆∪x∈Xω(x,f)⊆Ω(f).Fix(f) \subseteq Per(f) \subseteq Rec(f) \subseteq \bigcup\limits_{x \in X} \omega (x,f) \subseteq \Omega (f).

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.

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.

The next Theorem shows that Theorem 3 does not true for monotone maps on dendroids.

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]).

I. Construction of the dendroid X₁.

Let K be a Cantor set on the closed interval [0,1], a point p(12,12+i)∈ℂp({1 \over 2},{1 \over 2} + {\bf{i}}) \in {\mathbb C} . We set X1=∪e∈K[p,e].{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.

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;

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;

Now we set f₁ = g ○ τ : X₁ → X₁. By properties of maps τ and g, we get the following statements: 1)

f₁ is a homeomorphism and so f₁ is a monotone map;

Theorem 4 is proved.

I. Construction of the dendroid X₂.

We define a sequence {s_k}_k≥1 by the following way: (1)s0=0, sk=sk−1+2(2k−1), for k≥1.{s_0} = 0,\, {s_k} = {s_{k - 1}} + {2(2^k} - 1),\, {\rm{for}}\, \, k \ge 1.

We set (2)Ij=[12j;12j+i], for j∈{sk}k≥0.{I_j} = \left[ {{1 \over {{2^j}}};{1 \over {{2^j}}} + {\bf{i}}} \right],\, \, {\rm{for}}\, \, j \in {\{ {s_k}\} _{k \ge 0}}.

For any number n ∈ ℕ \ {s_k}_k≥1 there is a natural number k ≥ 0 such that s_k < n < s_k+1. It follows from (1) that for any k ≥ 0 every interval (s_k;s_k+1) contains 2^k+2 − 3 natural numbers. For every k ≥ 0 and any number 1 ≤ j ≤ 2^k+2 − 3 we define a vertical segmet I_sk+j by the following way: (3)Isk+j={[12sk+j;12sk+j+(1−j2k+1)i],if 1≤j≤2k+1−1;[12sk+j;12sk+j+j+2−2k+12k+1i],if 2k+1≤j≤2k+2−3.{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 I_n. Now we set X2=[0,1]∪[0,i]∪∪n=0∞In.{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.

Fig. 1

II. Construction of the map f₂ : X₂ → X₂.

We define a monotone map f₂ : X₂ → X₂ as follows: (i)

f₂(z) = z, if z ∈ [0,i];

III. Properties of f₂.

f₂ is a homeomorphism.

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 I_sk ∩ U(z) ≠ ∅. Then by (3) and (iii) for any j ≥ s_k and for any point x ∈ I_j(4)|Im f2(x)− Im x|≤12k+1,|{\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), (5)|Re f2(x)−Re x|=12j+1≤12k+1,|{\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 ≥ s_k and any point x ∈ I_j(6)|f2(x)−x|≤122(k+1)+122(k+1)=122k+1.|{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/2^k+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 12k1<d2.{1 \over {{2^{{k_1}}}}} < {d \over 2}. Now we take any natural number K ≥ k₁ such that I_sK ∩U(z) ≠ ∅. According to the choice of k₁ and (4) there is a natural number j ≥ 1 so that Im f2j(12sK+i)∈(Im z−d2,Im z+d2).{\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 f2sK+j(1+i)∈U(z)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.