Classification of the Regular Components of Two-Dimensional Inner Maps
We propose a topological classification of dynamical systems generated by two-dimensional inner maps on the fully invariant regular components of a wandering set with special attracting boundary (to within the topological conjugacy).
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Institute of Mathematics, NAS of Ukraine
2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508039715487744 |
|---|---|
| author | Vlasenko, V. F. Власенко, В. Ф. |
| author_facet | Vlasenko, V. F. Власенко, В. Ф. |
| author_sort | Vlasenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:24:29Z |
| description | We propose a topological classification of dynamical systems generated by two-dimensional inner maps on the fully invariant regular components of a wandering set with special attracting boundary (to within the topological conjugacy). |
| first_indexed | 2026-03-24T02:18:52Z |
| format | Article |
| fulltext |
UDC 513.83
I. Yu. Vlasenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
CLASSIFICATION OF THE REGULAR COMPONENTS
OF TWO-DIMENTIONAL INNER MAPS*
КЛАСИФIКАЦIЯ РЕГУЛЯРНИХ КОМПОНЕНТ
ДВОВИМIРНИХ ВНУТРIШНIХ ВIДОБРАЖЕНЬ
The topological classification of dynamical systems generated by two-dimentional inner maps on the fully invariant regular
components of a wandering set with a special attracting boundary up to the topological conjugacy is obtained.
Наведено топологiчну класифiкацiю динамiчних систем, породжених двовимiрними внутрiшнiми вiдображеннями
на iнварiантнiй регулярнiй компонентi зi спецiальною границею.
1. Introduction. Inner maps were introduced by Stoilov in [3]. Recall that a map is called open if
the image of an open set is an open set. A map is called discrete if a preimage of any point is a
discrete set (it consists of isolated points). A continuous open discrete map is called inner map. Note
that inner maps on compact surfaces have finite number of preimages.
The most noticeable representatives of inner maps are homeomorphisms and holomorphic maps.
In fact, an inner map of an oriented surface can be represented as a composition of a non-constant
analytical function and a homeomorphism due to Stoilov theorem [3]. It follows the class of inner
maps is much wider than holomorphic maps, for example, it includes all homeomorphisms. While
dynamics of homeomorphisms, diffeomorphisms and holomorphic maps has been studied deeply in
recent times the study of general inner maps with methods of dynamical systems theory just makes
its first steps.
The class of inner maps introduced here is based on the following observation. Consider a
holomorphic map zn. The foliation r = const obviously is an invariant foliation of this map. But
the important observation is the fact that the fibers of the foliation are dynamically invariant sets, as
for each point x the set ∪nf−n ◦ fn(x) is dense in a fiber. Thus, the map zn has a distinct invariant
foliation, which is preserved under topological conjugacy. Moreover, due to the Böttcher theorem
[1], this topologically invariant foliation naturally appears on the basin of attraction to infinity of the
complex polynomials of the Riemannian sphere.
This observation, though simple, seems to be a new one, not mentioned elsewhere in papers on
holomorphic dynamics.
We define the class of inner maps that have the same topologically invariant foliation on their
basin of attraction as those complex polynomials on the Riemannian sphere and call them the inner
maps with pseudolinear attracting isolated connected components.
Classification of dynamics on the attraction basin is done up to the topological conjugacy using
an analog of the Konrod – Reeb graph [2].
Theorem 1.1 (Main theorem). Restrictions of inner maps f, g on the fully invariant regular
components of wandering set with a fully invariant pseudolinear attracting isolated connected com-
* Supported by DKNII (M/377-2012) and DFFD (F40.1/009).
c© I. YU. VLASENKO, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1 41
42 I. YU. VLASENKO
ponent (Definition 2.4) are topologically equivalent if and only if they have the same degree and the
corresponding distinguishing graphs are equivalent.
2. Preliminary information. Let f : M →M be an inner map of a compact surface M. Let us
adapt to inner maps some classical dynamical systems definitions that are used for homeomorphisms.
Denote by O+(x) = {fn(x) | n ≥ 0} a forward trajectory of x, O−(x) = {f−n(x) | n ≥ 0} a
backward trajectory of x, and O(x) = {f−n ◦ fm(x) | n, m ≥ 0} a full trajectory of x. Let us call
the set O⊥(x) = {f−n ◦ fn(x) | n ≥ 0} a neutral section of trajectory.
Note that for the inner maps the situation is a bit different than in case of homeomorphisms,
where the definition of O⊥(x) has no sence as O⊥(x) = x. But for the inner maps the set O⊥(x)
usually contains an infinite number of points.
A point x is called wandering if there exists an open set U(x) such that ∀n ∈ Z, n 6= 0
fn (U(x))∩U(x) = ∅. A point which is not wandering is called nonwandering. The set of nonwan-
dering points of f is denoted by Ω(f).
Definition 2.1. A wandering point x is said to be regular if ∀ε > 0 ∃δ > 0 ∃N > 0 : ∀n > N
fn (Bδ(x)) ⊂ Bε(Ω) and f−n (Bδ(x)) ⊂ Bε(Ω), where Bε(X) is the ε-neighborhood of a set X.
A connected component of the set of regular wandering points is called a regular component of
wandering set. A component U is called periodic if there exists n such that fn(U) = U. When n = 1
a periodic component U is called invariant.
Note that for inner maps when U is invariant it does not follow that f−1(U) = U. Let us call an
invariant component fully invariant if f−1(U) = U.
Fully invariant components of inner maps are important because study of periodic components
can be essentially reduced to the study of fully invariant components.
Definition 2.2. An isolated connected component ∂0 of the boundary ∂U of a fully invariant
regular component U of wandering set is attracting if it has a strictly invariant neighborhood V (∂0)
(a neighborhood such that f(V ) ⊂ V ).
Let us properly define the subclass of inner maps studied here.
Definition 2.3. An attracting isolated connected component ∂0 of the boundary ∂U of a fully
invariant regular component U of wandering set is radially pseudolinear if it has strictly invariant
neighborhood V (∂0) such that there exists a regular compact invariant foliation of U ∩ V. Equiva-
lently, it can be defined as a foliation by a function Φ: V → [0, 1] such that Φ is regular in U ∩ V
and Φ(f(x)) = λΦ(x), λ > 0.
Definition 2.4. A radially pseudolinear attracting isolated connected component ∂0 of the bound-
ary ∂U of a fully invariant regular component U of wandering set is pseudolinear if
(A) ∀x ∈ U ∩ V the set O⊥(x) is dense in the foliation fiber of x;
(B) every fiber of the foliation has no more than one image of critical point.
3. Proof of the main theorem. Proof of the main theorem is split into 5 parts. In Subsection 3.1
we describe the topology of a regular component. Then in Subsection 3.2 we introduce a method to
select coordinates on a fundamental neighborhood, extend them on the whole regular component in
Subsection 3.3, use those coordinates to construct a distinguishing graph in Subsection 3.4 and build
a homeomorphism that provides a topological conjugacy in Subsection 3.5.
3.1. Topology of a fully invariant regular component. To prove intermediate Lemmas 3.1, 3.2
we need to adapt for inner maps a classic definition of a fundamental neighborhood.
Definition 3.1. A closed neighborhood Q ⊂ U is called fundamental neighborhood of U if
1) it is closure of its interior;
2) ∀x ∈ U O(x) ∩Q 6= ∅ (Q contains a representative of every full trajectory in U);
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
CLASSIFICATION OF THE REGULAR COMPONENTS OF TWO-DIMENTIONAL INNER MAPS 43
3) if x ∈ Q, x 6∈ ∂Q, then f(x) 6∈ Q.
Definition 3.2. Fundamental neighborhood Q is called saturated if for every x ∈ Q, Q contains
O⊥(x).
Lemma 3.1. Fundamental neighborhoods in the attracting neighborhood from the Definition 2.3
are homeomorphic to the ring.
Proof. Note that the attracting component of boundary is isolated, it means that the strictly in-
variant neighborhood is connected, so the foliation does not have holes. Otherwise other components
of boundary will be attracted to the attracting component, which will contradict isolatedness.
Since the boundary of strictly invariant neighborhood should also belong to the foliation and the
foliation is regular compact then all fibers are homeomorphic to the circle. Due to the regularity of
the foliation, the neighborhood does not contain critical points or their preimages. Hence, the map
between a fundamental neighborhood and its image is a regular covering. But a Mëbius string can
only cover a Mëbius string, handles can only cover handles, and if there exists any, there should be
infinitely many of them, as fundamental neighborhood has infinitely many images. It contradicts to
compactness. Since a fundamental neighborhood can’t have a handle or a Mëbius string, therefore Q
is homeomorphic to the ring.
Lemma 3.1 is proved.
Remark 3.1. In the Theorem 1.1 we consider a fully invariant pseudolinear attracting isolated
connected component of the boundary. Because it is fully invariant, then equiscalar lines of the
function Φ from the Definition 2.3 in the fundamental neighborhood Q from the Lemma 3.1 consist
of a single circle with the property that for its each point the circle also contains the whole set O⊥
of that point.
Lemma 3.2. A fully invariant regular component U of wandering set with a radially pseudolin-
ear attracting isolated connected component is either homeomorphic to the ring (have the only other
component of the boundary) in case it has no critical points or it has infinite number of boundary
components.
(a) (b)
Fig. 1. (a) Regular foliation, (b) foliation at the critical trajectory of degree 3.
Proof. By Lemma 3.1 all the fundamental neighborhoods in the attracting neighborhood from
the Definition 2.3 are homeomorphic to the ring. In absence of critical points all the fundamental
neighborhoods are homeomorphic, hence the whole regular component is homeomorphic to the ring.
In case when there are critical trajectories there are finite number of them due to compactness. By
Definition 2.3, the whole fully invariant neighborhood U ∩ V of the regular component U does not
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
44 I. YU. VLASENKO
contain critical points. Choose a fundamental neighborhood Q ⊂ U∩V. Note that Q is homeomorphic
to the ring according to the Lemma 3.1.
Consider the first preimage of Q that contain a critical point. It is enough to consider a single
critical point case, the deduction in the case of multiple critical points is the same.
According to the Stoilov’s “lemma on simple curve” [3], a critical point should distort the foliation
causing the fibers to intersect. In the image of critical point the foliation is regular, as on Fig. 1(a),
while in the critical point p and its preimages there is an intersection of n fibers, where n is the
degree of p. This situation is shown on Fig. 1(b).
P
0 /11 /2
1 /4
3 /4
1 /83 /8
5 /8 7 /8
1 /16
3 /165 /16
7 /16
9 /16
11 /16 13 /16
15 /16
(a) (b)
Fig. 2. (a) Wandering critical point, (b) numeration on O⊥(x).
Since fibers have dimension 1, they locally divide the surface M. As shown in the proof of
Lemma 3.1, the fibers of the fundamental neighborhood Q are circles. Then, the critical preimage
of a circle is a bouquet of circles. Every circle of the bouquet divide the regular component U.
Thus, preimage of Q has n + 1 components of boundary. This situation is shown on Fig. 2(a).
The set ∪∞k=−1fk(Q) also has n + 1 components of boundary because the rest of the fundamental
neighborhoods fk(Q), k ≥ 0, are homeomorphic to the ring. The further preimages of Q are disjoint
union of components of connectedness each of them is either homeomorphic to f−1(Q) or cover it.
It means that ∪∞k=−2fk(Q) will have not less than n2 +1 components of boundary, ∪∞k=−3fk(Q) will
have not less than n3 +1 components of boundary, and so on. As a consequence, U = ∪∞k=−∞fk(Q)
should have infinitely many components of boundary. If there are other critical points, the number of
boundary components just will grow faster, so the same reasoning is applicable too.
Lemma 3.2 is proved.
3.2. Coordinates on a fundamental neighborhood. Consider a saturated fundamental neighbor-
hood Q that is homeomorphic to the ring as in Lemma 3.1. Note that the boundary of Q consists of
foliation fibers. Choose an orientation on Q. As the points of O⊥(x) for a point x ∈ Q belong to the
same homeomorphic to a circle fiber (see Remark 3.1), they are naturally cyclically ordered. Assign
a point y ∈ f−n ◦ fn(x) ⊂ O⊥(x) a pair {k,mn}, where m is a degree of covering Q by f, n is
the iteration number as in f−n ◦ fn, the number k is a cyclic order of y among other points of the
preimage (f−n ◦ fn)(x) according to the orientation of the fiber. Note that 0 ≤ k < mn, 0 is for x,
1 is for the first another preimage of x under f−n ◦ fn in the direction of orientation. Let as write
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CLASSIFICATION OF THE REGULAR COMPONENTS OF TWO-DIMENTIONAL INNER MAPS 45
those pairs as fractions, for example, the pair {k,mn} is written as fraction
k
mn
. Assign the point x
a pair
0
1
. An example of the numeration for the degree 2 is shown on Fig. 2(b). Note that pairs that
denote the same numeric fraction
(
like
1
2
and
2
4
)
belong to the same point.
A fundamental neighborhood Q already has one family of coordinate lines due to the foliation.
The other family can be built as follows. Choose a point x ∈ ∂Q such that f(x) ∈ ∂Q. Then x
and f(x) are on different connected components of ∂Q which are fibers of the foliation. Connect
x and f(x) with a “transversal” to the foliation jordan curve γ that intersects each fiber in a single
point. Then the images f−n ◦ fn(γ) will not intersect with each other because for every x ∈ γ
the set O⊥(x) belongs to the fiber of x, but γ intersect fibers in a unique point. Then the set
O⊥(γ) = ∪∞n=0f
−n ◦ fn(γ) will yield an everywhere dense family of curves according to the
Definition 2.4 of pseudolinearity. Note that each image of the curve γ can be assigned a pair
k
mn
in
the same way as for the images of a point.
This lamination O⊥(γ) can be extended to foliation by continuity. For a point x ∈ Q \ O⊥(γ)
define µ(x) = ∩n≥0∆n(x), where ∆n(x) is closure of connected component of the setQ\f−n◦fn(γ)
that contains the point x. As an intersection of closed sets the set µ(x) has non-empty intersection
with every equiscalar line of foliation of Q. Show that µ(x) intersect each fiber in a unique point.
Suppose on the contrary that there exists a fiber ν such that it is intersected by µ(x) in 2 points
y1 and y2. Denote by z a point of intersection ν and γ. Then by construction the whole segment
[y1, y2] of the fiber ν belongs to µ(x). But it contradicts the fact that O⊥(z) is everywhere dense
in ν according to Definition 2.4. Hence µ(x) intersects each equiscalar line in a unique point. By
construction, µ(x) continuously depends on equiscalar lines foliation as µ(x) is majorated by images
of γ. As a consequence, it is also a jordan curve.
Thus, we constructed two families of curves on Q such that their fibers intersect each other in
a single point. Note that the foliation on equiscalar lines is a topological invariant of f, while the
second foliation is arbitrary up to the choice of γ.
3.3. Global coordinates on the regular component. Constructed above two families of curves
on Q generate two families of curves on the whole regular component U under the action of f and
f−1. Let us call a foliation of Φ the neutral foliation and the foliation induced by γ the timeline
foliation. By construction
they are invariant under the action of f ;
they are regular except in critical points and their preimages.
Remark 3.2. f acts as homeomorphism on non-critical fibers of the timeline foliation.
It follows from the fact that a fiber of timeline foliation intersects every fiber of neutral foliation,
and, hence, intersects each set O⊥ in no more then one point.
Note that neutral coordinates on neutral fibers are defined on the dense set O⊥(γ) and are
continuous by construction. They can be extended by continuity to a function ν : Q → S1 = [0, 1]
mod 1.
By choosing a function τ ′ : γ → [0, 1] and extending it to constant on the neutral fibers function
τ ′ : Q → [0, 1] every point p ∈ Q can be assigned unique coordinates (x = ν(p), y = τ(p)),
x ∈ [0, 1), y ∈ [0, 1].
If there are no critical points those coordinates can be extended to the coordinates on U (x, y),
x ∈ [0, 1), y ∈ R by the following rule. For a point p ∈ U a neutral coordinate x is determined by the
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
46 I. YU. VLASENKO
neutral coordinate of continuation of its timeline fiber, and the timeline coordinate y is determined
by equation y = y′ + n, where y′ = τ(fn(p)) is timeline coordinate of fn(p) ∈ Q. Call them
γ-coordinates.
Lemma 3.3. Two regular components U1 and U2 of wandering sets of inner maps f and g of the
same degree with a pseudolinear attracting isolated connected components of the boundary without
critical points are topologically conjugate.
Proof. Choose arbitrarily saturated fundamental neighborhoods Q1 and Q2 of U1 and U2 and
transversal curves γ1 ⊂ Q1 and γ2 ⊂ Q2 to neutral foliations in Q1 and Q2. Construction above
extends them to the two pairs of foliations in Q1 and Q2.
Choose a homeomorphism h′γ : γ1 → γ2. Continue h′γ over the timeline foliation using maps
(g−k ◦ gl)−1 ◦ h′γ ◦ f−k ◦ f l, k, l ≥ 0. Note that branches of those maps are homeomorphisms
when restricted on a single fiber (see Remark 3.2) so they have inverse homeomorphisms which are
denoted here as (g−k ◦ gl)−1. Denote the obtained map by ĥγ . Since f and g preserve their neutral
foliations, ĥγ maps points that belong to the same fiber of the neutral foliation of f to the points on
the same fiber of the neutral foliation of g. Because f and g have the same map degree they act the
same way on the points that have the same neutral coordinates. It means that ĥγ induces not only
mapping of timeline foliations, but also a mapping of neutral foliations.
It can be extended by continuity to a bijection hγ between U1 and U2. This bijection is continuous,
since foliations are continuous. Also hγ is an open map because in every point it maps its base of
topology built from foliation boxes to the corresponding base of topology generated by foliations in
the image. It shows that hγ is a homeomorphism. By construction, hγ ◦ f = g ◦ hγ , it means that f
and g are topologically conjugate.
Note that if one choose h′γ to map identical timeline γ-coordinates on U1 and U2 then hγ in
corresponding γ-coordinates on U1 and U2 will be nothing but Id .
Lemma 3.3 is proved.
3.4. Distinguishing graph. As shown in Lemma 3.3 in case when there are no critical points the
topological classification is simple. The only essential invariant is the map degree.
In case when there are critical points the symmetry of points breaks. Topological conjugacy
distinguish among critical and non-critical points, their preimages and foliation fibers. To deal with
this extra topological invariants we need to build a distinguishing graph which is technically a
compact encoding of some labelled Konrod – Reeb graph of a specially chosen part of the foliation
function, see [2].
Choose the fundamental neighborhood to be the neighborhood between the boundary of the first
and second images of the first critical level of the invariant neutral foliation. Note that due to the
choice of the fundamental neighborhood and the condition B) from Definition 2.4 any fiber of the
neutral foliation in the chosen fundamental neighborhood has no more than one image of a critical
point and there are finitely many fibers that have an image of a critical point. Choose the line γ
that generates the timeline foliation to intersect each fiber with a critical point in that critical point,
including the boundary fibers of the fundamental neighborhood. In absence of critical points the
points of regular component can be uniquely identified by a pair of coordinates on its timeline and
neutral fibers (γ-coordinates introduced above). But the critical points make the foliation singular so
that lines of the timeline foliation intersect in critical points and their preimages. Also preimages of
the fundamental neighborhood divide onto components such that internals of those components are
mutually disconnected. It follows that timeline γ-coordinates can be defined in the same way as in
the absence of critical points, but neutral γ-coordinates can’t.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
CLASSIFICATION OF THE REGULAR COMPONENTS OF TWO-DIMENTIONAL INNER MAPS 47
In that case introduce neutral γ-coordinates that are local to a component of a preimage of the
fundamental neighborhood Q. Consider the fiber γ̂ of timeline foliation that contains the original
curve γ ∈ Q. This fiber branches in critical points and their preimages simultaneously together
with dividing of preimages of the fundamental neighborhood Q onto components so than there is
a unique branch of the singular fiber γ̂ that goes into internals of each component. Now construct
neutral coordinates for every component using this branch of γ̂ as local origin
0
1
. At that points at
the component boundary might get different neutral coordinates from components the point borders
with. To avoid the ambiguity let us always assume that the component of the boundary further from
attractor should belong to the component and inherit its numeration. From that the critical points and
their preimages will get the neutral coordinate
0
1
.
Also, those coordinates are not unique: there are finitely many components having the same
coordinates. To avoid this ambiguity component of a preimage should be counted for every n such
that every point of U could be identified with a triplet (x, y, C), where x is a local neutral coordinate,
y is a global timeline coordinate, and C is a component number.
To enumerate the components choose an orientation on the boundary of the original fundamental
neighborhood Q. Since the regular component is oriented, this orientation induces an orientation on
the preimages of the boundary. Starting from a point on the original curve γ and move on the singular
fiber γ̂ there is a unique shortest path to walk over all components of preimage of the border of Q
moving on the preimages according to the orientation and moving according to the orientation from
one component to another on γ̂. That path cyclically orders components. Number them according to
that order so the first component met on that path will get the number 1. This numeration is related
to the first critical point.
The other critical points further subdivide some of those components, in such a way that those
subcomponents share the same regular component of the singular neutral foliation of the previous
critical point. so in that case it is natural to create a compound number by assigning such a component
the sequence of numbers each related to the critical neutral fiber of corresponding critical point. Call
the obtained compound number C = {k1, . . . , kl} a component number.
By construction critical points are identified by a triplet
(
0
1
, n+ y′, C
)
, which does not depend
on the choice of γ. It only has compound component number depending on choice of orientation and
timeline coordinate y having invariant integer part n and a fraction part y′ depending on the choice
of the timeline function τ. However, the relative order of images and preimages of critical neutral
fibers does not depend on the choice of τ.
Project the critical points on the semi-interval [0, 1) using their fractional parts y′ of their timeline
coordinates and label them with pairs (d, n, CC), where d is a local degree of the critical point, CC
is an unordered pair of component numbers with different choices of orientation in Q. Note that the
first critical point gets by construction the label (. . . , 0, {1}{1}).
Definition 3.3. Two labels are equal if their components are equal: the first and second compo-
nents are equal as numbers, the third components are equal as unordered pairs of vectors.
Definition 3.4. Call the semi-interval [0, 1) either without labelled points or with labelled points
such that 0 is labelled and 0’s label looks like (. . . , 0, {1}{1}) a distinguishing graph.
Definition 3.5. Two distinguishing graphs are equivalent if there exists a homeomorphism of the
semi-interval [0, 1) such that the labels of images and preimages are equal.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
48 I. YU. VLASENKO
3.5. A conjugating homeomorphism. To finish the proof of the main theorem we need to build
a homeomorphism that provide the conjugacy of inner maps on their regular components with equiv-
alent distinguishing graphs.
The construction is similar to that in the proof of the Lemma 3.3, except we have not global
but local coordinate families in each component of preimage of the fundamental neighborhood that
are built is Subsection 3.4. By definition the equivalence of the distinguishing graphs guarantees
one-to-one correspondence between adjacent components. As in Lemma 3.3, it follows that a map
written in that coordinates as Id is the desired homeomorphism.
1. Milnor J. Dynamics in one complex variable. Introductory lectures. – Weisbaden: Vieweg Verlag, 2000. – vii+257 p.
2. Sharko V. V. About Kronrod – Reeb graph of a function on a manifold // Meth. Funct. Anal. and Top. – 2006. – 12,
№ 4. – P. 389 – 396.
3. Stoı̈low S. Leçon sur les principes topologiques de la theorie des fonctions analytiques. – 2 ed. – Paris: Gauthier-Villars,
1956. – 194 p.
4. Trokhymchuk Yu. Yu. Differentiating, inner mappings and analyticity criteria // Proc. Inst. Math. – 2008. – 70. – 538 p.
Received 13.11.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 1
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| spelling | umjimathkievua-article-21092019-12-05T10:24:29Z Classification of the Regular Components of Two-Dimensional Inner Maps Класифікація регулярних компонент двовимірних внутрішніхх відображень Vlasenko, V. F. Власенко, В. Ф. We propose a topological classification of dynamical systems generated by two-dimensional inner maps on the fully invariant regular components of a wandering set with special attracting boundary (to within the topological conjugacy). Наведено топологічну класифікацію динамічних систем, породжених двовимірними внутрішніми відображеннями на інваріантній регулярній компоненті зі спеціальною границею. Institute of Mathematics, NAS of Ukraine 2014-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2109 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 1 (2014); 41–48 Український математичний журнал; Том 66 № 1 (2014); 41–48 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2109/1220 https://umj.imath.kiev.ua/index.php/umj/article/view/2109/1221 Copyright (c) 2014 Vlasenko V. F. |
| spellingShingle | Vlasenko, V. F. Власенко, В. Ф. Classification of the Regular Components of Two-Dimensional Inner Maps |
| title | Classification of the Regular Components of Two-Dimensional Inner Maps |
| title_alt | Класифікація регулярних компонент двовимірних внутрішніхх відображень |
| title_full | Classification of the Regular Components of Two-Dimensional Inner Maps |
| title_fullStr | Classification of the Regular Components of Two-Dimensional Inner Maps |
| title_full_unstemmed | Classification of the Regular Components of Two-Dimensional Inner Maps |
| title_short | Classification of the Regular Components of Two-Dimensional Inner Maps |
| title_sort | classification of the regular components of two-dimensional inner maps |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2109 |
| work_keys_str_mv | AT vlasenkovf classificationoftheregularcomponentsoftwodimensionalinnermaps AT vlasenkovf classificationoftheregularcomponentsoftwodimensionalinnermaps AT vlasenkovf klasifíkacíâregulârnihkomponentdvovimírnihvnutríšníhhvídobraženʹ AT vlasenkovf klasifíkacíâregulârnihkomponentdvovimírnihvnutríšníhhvídobraženʹ |