On Limit Sets of Monotone Maps on Dendroids

E.N. Makhrova 1
  • 1 Lobachevsky State University of Nizhny Novgorod, , Russia
E.N. Makhrova

Abstract

Let X be a dendrite, f : XX be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point xX has the next properties:

  1. (1)ω(x,f)Per(f)¯ , where Per( f ) is the set of periodic points of f ;
  2. (2)ω(x, f ) is either a periodic orbit or a minimal Cantor set.

In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that

  1. (3)Ω(f)=Per(f)¯ , where Ω( f ) is the set of non-wandering points of f.

The aim of this note is to show that the above results – do not hold for monotone maps on dendroids.

1 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 = AB, then AB 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,yX 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 pX is called to be

  1. an end point of X if the set X \ {p} is connected;
  2. 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 pX are at most countable (see [1, §51]). These statements are not true for dendroids.

Let f : XX be a continuous map of a dendroid X. ω-limit set of a point xX is the set
ω(x,f)={zX:nj,nj,limjfnj(x)=z}.
Definition 2

A point xX is said to be

  1. 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 ;
  2. a recurrent point of f if xω(x, f );
  3. 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)Per(f)Rec(f)xXω(x,f)Ω(f).
Definition 3

[1, §46] Let f : XX be a continuous map of a dendroid X. A map f is said to be monotone if for any connected subset Cf (X), f−1(C) is connected.

Let f : XX be a monotone map. Denote by f n the n-iterate of f ; that is, f0 = identity and f n = ff 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 : DD be a monotone map of a dendrite D. Then for any point xD,ω(x,f)Per(f)¯ .

Theorem 2

[4] Let f : DD be a monotone map of a dendrite D. ThenΩ(f)=Per(f)¯ .

Theorem 3

[5] Let f : DD be a monotone map of a dendrite D. Then for any point xD, ω(x, f ) is either a periodic orbit or a minimal Cantor set.

In the note we show that Theorems 13 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 : X1X1such that

  1. (4.1)ω(x,f1)Per(f1)¯for some point xX1;
  2. (4.2)Ω(f1)Per(f1)¯ .

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 : X2X2such that for some point xX2, 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]).

2 Proof of Theorem 4

I. Construction of the dendroid X1.

Let K be a Cantor set on the closed interval [0,1], a point p(12,12+i) . We set
X1=eK[p,e].
Note that X1 is a dendroid which is not a locally connected continuum in any point xX1 \ {p}.

II. Construction of the map f1 : X1X1.

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
dΣ((k1,k2,),(j1,j2,))=i=1+δ(ki,ji)2,
where δ (ki, ji) = 1, if kiji and δ (ki, ji) = 0, if ki = ji. The addition in Σ is defined as follows:
(k1,k2,)+(j1,j2,)=(l1,l2,),
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,
σ((j1,j2,j3,))=(j1,j2,j3,)+(1,0,0,).
Lemma 6

[12], [13]

  1. 1.Σ is a Cantor set;
  2. 2.σ : Σ → Σ is a homeomorphism;
  3. 3.Per(σ) = ∅;
  4. 4.Rec(σ) = Σ.

To define a map f1 : X1X1 we need two auxiliary maps.

1. Let h : K → Σ be any homeomorphism. We define a map τ : X1X1 as follows: τ : [p,e] → [p,h−1σh(e)] be a linear homeomorphism so that τ(p) = p , τ(e) = h−1σh(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. xRec(τ) \ Per(τ) for any point xX1 \ {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 : X1X1 by the following way: for any point eK

g : [p,e] → [p,e] be a homeomorphism such that g(x) = ϕ−1x2ϕ(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 eK and an arbitrary point x ∈ (p,e], ω(x,g) = {e}.

Now we set f1 = gτ : X1X1. By properties of maps τ and g, we get the following statements:

  1. 1)f1 is a homeomorphism and so f1 is a monotone map;
  2. 2)Per( f1) = Fix( f1) = {p};
  3. 3)for any point xX1 \{p}, ω(x, f1) is a minimal Cantor set K, that is ω(x, f1) = K. Hence, ω(x,f1)Per(f1)¯ .
  4. 4)Ω( f1) = {p} ∪ K. So Ω(f1)Per(f1)¯ .

Theorem 4 is proved.

3 Proof of Theorem 5

I. Construction of the dendroid X2.

We define a sequence {sk}k≥1 by the following way:
s0=0,sk=sk1+2(2k1),fork1.
We set
Ij=[12j;12j+i],forj{sk}k0.
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:
Isk+j={[12sk+j;12sk+j+(1j2k+1)i],if1j2k+11;[12sk+j;12sk+j+j+22k+12k+1i],if2k+1j2k+23.
It follows from (2) and (3), that for any number n ∈ ℕ ∪ {0} we defined a segment In. Now we set
X2=[0,1][0,i]n=0In.
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
Fig. 1

Dendroid homeomorphic to X2.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00056

II. Construction of the map f2 : X2X2.

We define a monotone map f2 : X2X2 as follows:

  1. (i)f2(z) = z, if z ∈ [0,i];
  2. (ii)f2(z) = z/2, if z ∈ [0,1];
  3. (iii)f2 : IjIj+1 be a linear homeomorphism such that f2(Ij) = Ij+1 for any number j ≥ 0.

III. Properties of f2.

  1. f2 is a homeomorphism.
  2. Per( f2) = Fix( f2) = [0,i].
  3. We show that f2 is a continuous map.

It is evident that f2 is a continuous map in any point zX2 \ [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 IskU(z) ≠ ∅. Then by (3) and (iii) for any jsk and for any point xIj
|Imf2(x)Imx|12k+1,
where Im* is the imaginary part of a complex number *. By (ii) and (iii),
|Ref2(x)Rex|=12j+112k+1,
where Re* is a the real part of a complex number *.
It follows from (4) and (5) that for any jsk and any point xIj
|f2(x)x|122(k+1)+122(k+1)=122k+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
12k1<d2.
Now we take any natural number Kk1 such that IsKU(z) ≠ ∅. According to the choice of k1 and (4) there is a natural number j ≥ 1 so that
Imf2j(12sK+i)(Imzd2,Imz+d2).
It follows from here that f2sK+j(1+i)U(z) . So, zω(1 + i, f2).

Thus, ω(1 + i, f2) = [0,i] = Fix( f2). Theorem 5 is proved.

References

  • [1]

    Kuratovski K. (1968) Topology, vol.2 (New York: Academic Press).

  • [2]

    Nadler S. B. (1992) Continuum Theory: An Introduction (Monographs and Textbooks in Pure and Applied Mathematics, 158). Marcel Dekker, New York.

  • [3]

    Naghmouchi I. (2011) Dynamics of monotone graph, dendrite and dendroid maps International Journal of Bifurcation & Chaos21 3205–3215.

    • Crossref
    • Export Citation
  • [4]

    Makhrova E. N., Vaniukova K. S. (2016 ) On the set of non-wandering of monotone maps on local dendrites Journal of Physics: Conference Series692, 012012.

  • [5]

    Naghmouchi I. (2012) Dynamical properties of monotone dendrite maps Topology and its Applications159 144–149.

    • Crossref
    • Export Citation
  • [6]

    Balibrea F., García Guirao J.L., Muñoz Casado J.I. (2001) Description of ω-limit sets of a triangular map on I2 Far East J. Dyn. Syst.3 87–101.

  • [7]

    Balibrea F., García Guirao J.L., Muñoz Casado J.I. (2002) A triangular map on I2 whose ω-limit sets are all compact interval of 0 × I Discrete Contin. Dyn. Syst.8 983–994.

    • Crossref
    • Export Citation
  • [8]

    Efremova L.S. (2017) Dynamics of skew products of interval maps Russian Math. Surveys72 101–178.

    • Crossref
    • Export Citation
  • [9]

    Efremova L. S. (2010) Differential properties and attracting sets of a simplest skew product of interval maps Sb. Math.201 873–907.

    • Crossref
    • Export Citation
  • [10]

    Kočen Z. (1999) The problem of classification of triangular maps with zero topological entropy Ann. Math. Sil.13 181–192.

  • [11]

    Kolyada S.F. (1992) On dynamics of triangular maps of the square Ergodic Theory Dynam. Systems12 749–768.

    • Crossref
    • Export Citation
  • [12]

    Buescu J., Stewart I. (1995) Liapunov stability and adding machines Ergodic Theory and Dynamical Systems15 271–290.

    • Crossref
    • Export Citation
  • [13]

    Block L., Keesling J. (2004) A characterization of adding machine maps Topology and its Applications140 151–161.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Kuratovski K. (1968) Topology, vol.2 (New York: Academic Press).

  • [2]

    Nadler S. B. (1992) Continuum Theory: An Introduction (Monographs and Textbooks in Pure and Applied Mathematics, 158). Marcel Dekker, New York.

  • [3]

    Naghmouchi I. (2011) Dynamics of monotone graph, dendrite and dendroid maps International Journal of Bifurcation & Chaos21 3205–3215.

    • Crossref
    • Export Citation
  • [4]

    Makhrova E. N., Vaniukova K. S. (2016 ) On the set of non-wandering of monotone maps on local dendrites Journal of Physics: Conference Series692, 012012.

  • [5]

    Naghmouchi I. (2012) Dynamical properties of monotone dendrite maps Topology and its Applications159 144–149.

    • Crossref
    • Export Citation
  • [6]

    Balibrea F., García Guirao J.L., Muñoz Casado J.I. (2001) Description of ω-limit sets of a triangular map on I2 Far East J. Dyn. Syst.3 87–101.

  • [7]

    Balibrea F., García Guirao J.L., Muñoz Casado J.I. (2002) A triangular map on I2 whose ω-limit sets are all compact interval of 0 × I Discrete Contin. Dyn. Syst.8 983–994.

    • Crossref
    • Export Citation
  • [8]

    Efremova L.S. (2017) Dynamics of skew products of interval maps Russian Math. Surveys72 101–178.

    • Crossref
    • Export Citation
  • [9]

    Efremova L. S. (2010) Differential properties and attracting sets of a simplest skew product of interval maps Sb. Math.201 873–907.

    • Crossref
    • Export Citation
  • [10]

    Kočen Z. (1999) The problem of classification of triangular maps with zero topological entropy Ann. Math. Sil.13 181–192.

  • [11]

    Kolyada S.F. (1992) On dynamics of triangular maps of the square Ergodic Theory Dynam. Systems12 749–768.

    • Crossref
    • Export Citation
  • [12]

    Buescu J., Stewart I. (1995) Liapunov stability and adding machines Ergodic Theory and Dynamical Systems15 271–290.

    • Crossref
    • Export Citation
  • [13]

    Block L., Keesling J. (2004) A characterization of adding machine maps Topology and its Applications140 151–161.

    • Crossref
    • Export Citation
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