Recently authors the Birkhoﬀ curves additive

theory has been constructed [1, 2, 3]. The Birkhoﬀ

curves were researched to be invariant continua

with respect to the dissipative dynamic systems

ψ:N→Diﬀ(E2) acting on Euclidean plane de-

ﬁned by the formula

P(x, y) + iQ(x, y)def

7→ x+iy (1)

at iterations. Also the authors had constructed

the Birkhoﬀ curves at presence of prime or com-

plex equilibrium point [4].

Inter alia there subsists invariant set Λ (i. e.

Λdef

== ψk(Λ) for all k∈Z) on the plane, such that

a homomorphism action

ψ:Z→Diﬀ(Λ)

has been deﬁned by the formula (1).

The Birkhoﬀ curve separates the plane and

one turns out to be indecomposable continuum

(atom) (even if Birkhoﬀ curve separates the plane

on two regions). Then the invariant set Λ with

respect to dissipative action ψis constituted to

be plane connected compact.

The colouring invariant regions problem has

been solved author in [5, 6]. Also this problem

solved in [9, 10, 11]. However the colouring prob-

lem every time had only empiric solution.

Suppose Υ be a Birkhoﬀ curve being two or more

invariant regions common boundary on the plane

and Υ ⊂Λ. If action ψturns out to be dissipative

then addition E2\Λ be only semi-invariant set.

Now suppose that the Birkhoﬀ curve Υ is ei-

ther the boundary of Λ, or Υ separates Λ into ν

invariant regions Gi,i∈1, ν, being their common

boundary. Then the following equalities

Υdef

== Fr G1=. . . = Fr Gν= Fr(E2\Λ) def

== Fr G0

are faithful.

Wada Basins Colourings and Ergodicity

PRASKWYA D. SEROWA1, DMITRY W. SEROW2

1State University of Film and Television of Saint Peterburg

13 Pravda Str. 191119 Saint Peterburg (Hero City Leningrad). RUSSIA

2Dynamic Systems National Research Centre | SEQUOIA,

188300 Military Glory City Gatchina, RUSSIA

Abstract: - The stable Birkhoff curve ergodic properties in dissipative situation have been

discussed. The points of every regions have been coloured with their own colour, such that their

common boundary would turns out to be white at the colours adding together. Quantitative

conditions for colouring regions for boundary discolouration have been obtained. The prime

example of the dynamic system action with fixed B- saddle and Birkhoff curve being four

regions common boundary has been constructed. There subsist the Birkhoff curves different

topologic types. For every Birkhoff curves can constructed to be two or more vortex streets

with periodic structure.The regions colouring turns out to be the topological classification

instrument.

Keywords: -Wada basins, Wada property, indecomposable continuum (atom), composanta,

rotation number, γ-density, Schnirelmann density, ergodic theorem, PostScript

Received: March 5, 2024. Revised: September 8, 2024. Accepted: October 5, 2024. Published: November 5, 2024.

1. General Remarks

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Remark 1 The Birkhoﬀ curve Υis constituted

to be (atom)and therefore one consists of only

tops of 1-umbrellas, because the oscillations in ev-

ery point of atom is equal to its diameter (see [1,

13] and e. g. [12], vol. II or [14]).

Therefore all tops of 1-umbrellas containing in

Birkhoﬀ curve turn out to be irreducible points.

It is clear that the Birkhoﬀ curve is not a mani-

fold, even if it turns out to be two regions com-

mon boundary.

The accessible point from region Gi,i∈1, ν

is said to be point ◦

qi∈Υ, such that there exists

arc γiwith end being ◦

qiand γi\ {◦

qi} ∈ Gi. The

Wada basins

◦

Giare called to be invariant regions

Giand set of all accessible points from Giunion

for all i∈1, ν, if ν>2 (see also e. g. [3]). The

Wada ocean

◦

G0is said to be invariant region G0

and set of all accessible points from G0union.

In this connotation, the following statements

Υis the Birkhoﬀ curve and ◦

Gi∩◦

Gj= Ø for all

i, j ∈0, ν,ν>1and i6=jare equivalent (see

[15]). However there are no reasons to assume

that Υ does not contain inaccessible points from

any of the invariant regions.

For simplicity of further narration suppose

that Birkhoﬀ curve turns out to be ω-limit conti-

nuum for dissipative action ψ. Then there subsist

νﬁxed or periodic unstable antisaddles contained

into invariant regions with compact closure. This

means that the points of every trajectory accu-

mulate close enough to Υ, with the exception,

perhaps, of the ﬁxed or periodic unstable antisad-

dles. Moreover every trajectory close enough to

Υ everywhere dense, with the exception, perhaps,

of the ﬁxed or periodic points (for this occasion,

see [1]). This dynamic systems property Poincar´e

has been called to be ergodic (see e. g. [7]). It

should be noted that the action ψis dissipative,

but the invariant regions are incompressible.

Now let us colouring the Wada ocean and ev-

ery Wada basin in its own colour deﬁned, for in-

stance, by the triple of positive numbers not ex-

ceeding one (ri,gi, bi), i∈0, ν so, that

06i6ν

(ri,gi, bi)≡(1,1,1).(2)

Then what should be the ¡¡colouring density¿¿

of every invariant region (or Wada ocean and

basins) for their common border turns out to be

white; or that too it has been deﬁned by triple

of ones (1,1,1) in the neighborhood of every the

boundary point?

A successful colouring has been obtained at the

dynamic system constructing with two invariant

centrally symmetric Wada basins [6]. However,

this colouring turned out intuitive or empirical.

First us deﬁne what is the colouring density.

Suppose u1, u2, . . . , uνare unstable antisaddles

with their neighbourhoods

U(u1), U(u2), . . . , U(uν),

such that U(ui)⊂Gifor all i∈1, ν. In ad-

dition, let us choose some point u0in G0with

neighbourhood U(u0), such that U(u0)⊂G0.

For deﬁniteness one can suppose that all the

neighbourhoods U(ui) appear identical open small

squares. Then the coloured points number #U(ui)

is called to be the region Gicolouring density.

So, coloured the open small squares of the

colouring densities

#U(u0),#U(u1), . . . , #U(uν)

have been determined.

Due to the fact that continuum Υ appears

stable boundary there are true the following

Proposition 1 For any neighbourhood U(ξ)of

the point ξ∈Υthere exists number k∈N, such

that for the forms of the map

ψk(U(u0)), ψk(U(u1)), . . . , ψk(U(uν))

the intersections ψk(U(ui))∩U(ξ)are non-empty

for all i∈0, ν.

Corollary 1 For a suﬃciently large k∈Nthe

inequality #(ψk(U(ui)) ∩U(ξ)) >1is faithful.

Suppose Giis appeared to be invariant region

with respect to the action ψcontaining neigh-

bourhood U(ui) of periodic point ui. Now assume

U(ui) contains either #U(ui) colouring points or

the only colouring point ci6=ui, if i∈1, ν. In

these ways, one can start colouring the Wada

basins. However for G0one can colourize point

2. Wada Basins Colouring

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u0=c0and neighbourhood U(c0) containing ei-

ther #U(c0) colouring points or the only colour-

ing point c0. In these ways, one can start colour-

ing the Wada ocean.

In the sequel, the basins and ocean can be col-

ored in the following two ways, either the forms

of the neighbourhoods U(ui) at iterations

U(ui), ψ1(U(ui)), . . . , ψK(U(ui)), . . .

with #U(ui) colouring points colourize the in-

variant regions or semi-trajectories O+(ci) colour-

ize the invariant regions for all i∈0, ν. How-

ever the colours regions satisfy to condition (2)

and the colouring points numbers ratios or the

lengths of colourized semi-trajectories ratios must

appeared to be such that the boundary contin-

uum (Birkhoﬀ curve) turns out to be white.

Remark 2 The condition (2) turns out to be sca-

lar. Indeed one can apply ¡¡grayscales¿¿ Sifor

colouring as follows P

06i6ν

Si≡1. It means that

for every region colouring by the only (scalar)pa-

rameter is determined.

It is quite natural to consider numerical topo-

logical or metric invariants of the region, for in-

stance the rotation number or γ-density [2], as a

colouring parameters.

The additive rotation number theory has been

constructed in application to the Birkhoﬀ curve

(see [1]). The theory is interesting but one is

trivial applied to Jordan curve.

Birkhoﬀ curves turn out to be nonwandering

continua Υ with respect to the dissipative dy-

namic systems ψacting on the plane, such that

Υ separates the plane. Its appear in diﬀerent dy-

namic situations.

For every invariant region Gl,l∈0, ν and for

Wada basin ◦

Githe iterations numbers sequence

Kl:k1= 1, k2, k3, . . . has been deﬁned. Rota-

tion number for ◦

Gibeing Schnirelmann density

for Klis deﬁned to be following formula

σKl

def

== inf

K∈N

#({1, K} ∩ Kl)

K(3)

(compare with the deﬁnition e. g. from [8]). Such

a way the set

{σK0, σK1, . . . , σKl, . . . , σKν}

turns out to be topological invariant for bound-

ary Υ. The Schnirelmann densities for Klcorre-

sponding to ◦

Giturn out to be irrational numbers.

It means that that accessible point trajectory is

everywhere dense on Υ (the theory details, see

in [1]).

Proposition 2 If rotation numbers of basins are

equal then the colouring rates of their common

boundary are equal.

PROOF. Indeed, if the basins ◦

Giand ◦

Gj,i, j ∈

0, ν, such that their rotation numbers are equal,

i. e. σKi=σKjfor i6=j. Then there exists

number ord Ki= ord Kj

def

== η, such that

σ(η⊕Ki) = σ(η⊕Kj)≡1,

or that too

η⊕Ki=η⊕Kj≡N.

It means that colouring rate of region Giis equal

to colouring rate of region Gjut

Theorem 1 The diﬀerence in the rotation num-

bers of regions (basins)means the colouring rate

diﬀerence of their common boundary.

PROOF. Suppose ◦

Giand ◦

Gj,i, j ∈0, ν, such

that their rotation numbers are diﬀerent, i. e. for

instance σKi> σKjfor i6=j. Then there sub-

sist two numbers ord Ki>ord Kj(called the order

sequence), such that

σ(ord Ki⊕Ki) = σ(ord Kj⊕Kj)≡1,

or that too

ord Ki⊕Ki= ord Kj⊕Kj≡N.

It means that colouring rate of region Gjdoes

not exceed of colouring rate of region Gi. In the

event that ord Ki= ord Kjlet us cross it out,

for instance, even elements from the sequences Ki

and Kj. Thus new sequences K0

iand K0

jconsisting

of odd elements of the sequences Kiand Kjare

3. Densities and Colourings

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obtained. Then σK0

i> σK0

jand there subsist two

numbers ord K0

i>ord K0

j, such that

σ(ord K0

i⊕K0

i) = σ(ord K0

j⊕K0

j)≡1.

Next, the process of crossing out even elements of

sequences is repeated until for new two sequences

σK00

i> σK00

jtheir orders will satisfy to the inequal-

ity ord Ki>ord Kjut

Remark 3 Inter alia crossing out scheme can

be anything, if only it was the same for both se-

quences Kiand Kj.

Moreover if Υ turns out to be Birkhoﬀ curve

then sequence Klcontains zero Schnirelmann den-

sity additive basis Blfor all ◦

Gland l∈0, ν. In

order to compare sequences zero Schnirelmann

density the γ-density has been applied deﬁned by

the formula

γBl

def

== inf

K∈N

log(1 + #(1, K ∩Bl))

log(K+ 1) (4)

The γ-density theory in detail has been narrated

in [2, 3].

Due to the sequences B0,B1, . . . , Bνturn

out to be additive bases then for every l∈0, ν

there exist numbers

ord B0,ord B1, . . . , ord Bν,

such that σ(ord Bl⊕Bl)≡1.

Proposition 3 If γ-densities deﬁned for regions

(basins)are equal then the colouring rates of their

common boundary are equal.

PROOF exactly repeats the proof of the propo-

sition (2) ut

Theorem 2 The diﬀerence in γ-densities deﬁned

for regions (basins)means the colouring rate dif-

ference of their common boundary.

PROOF. Suppose ◦

Giand ◦

Gj,i, j ∈0, ν, such

that their γ-densities are diﬀerent, i. e. for in-

stance γBi> γBjfor i6=j. Then there exist

two numbers ord Bi>ord Bj, such that

σ(ord Bi⊕Bi) = σ(ord Bj⊕Bj)≡1,

or that too

ord Bi⊕Bi= ord Bj⊕Bj≡N.

It means that colouring rate of region Gjdoes

not exceed of colouring rate of region Gisame as

for rotation numbers. In the event that ord Bi=

ord Bjat the crossing out even elements from the

sequences Kiand Kjthe sequences K0

iand K0

j, con-

taining B0

i⊂Biand B0

j⊂BJrespectively, are

obtained. Thus new bases B0

iand B0

jconsisting

of odd elements of the sequences Kiand Kjhave

been obtained. Therefore γB0

i> γ B0

jand there

subsist two numbers ord B0

i>ord B0

j, such that

σ(ord B0

i⊕B0

i) = σ(ord B0

j⊕B0

j)≡1.

Next, the process of crossing out even elements of

sequences is repeated until for new two sequences

σK00

i> σK00

jtheir orders will satisfy the inequality

ord B00

i>ord B00

jfor both bases B00

i⊂K00

iand

B00

j⊂K00

jrespectively ut

Thus, the colouring rates are only provided by

the existence of additive bases zero Schnirelmann

density Blwith γ-densities values for all diﬀerent

invariant regions. However the colouring rates

are determined by corresponding to these regions

the rotation numbers.

Let us place

#U(u0),#U(u1), . . . , #U(ul), . . . , #U(uν)

colouring points contained in U(ul)⊂Glfor ev-

ery invariant region respectively, and let us build

for every colouring point ζfrom neighbourhood

U(ul) the sequence Kl(ζ) on the following scheme

described in [1]:

1◦suppose that an arbitrary ray Blis selected

with its beginning at the point ul, so that

one sets the direction angle ϕ; then for ev-

ery point the direction with respect to the

point ulbeing equal arg ξis deﬁned;

2◦a forms of all colouring points Klcontained

in U(ul)⊂Glis separated on the ﬁnite dis-

joint classes number

K(1)

l,K(2)

l, . . . , K(η)

l(5)

as follows:

(1) ζ∈K(1)

lif arg(ψ1(ζ)) >arg ξ;

4. Invariant Regions Colouring

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(2) ζ∈K(2)

lif ξ∈Kl\K(1)

land

arg(ψ2(ζ))) >arg ζ;

(3) ζ∈K(3)

lif ζ∈Kl\K(1)

l\K(2)

land

arg(ψ3(ξ))) >arg ζ;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(η)ζ∈K(η)

lif ζ∈Kl\K(1)

l\. . . \K(η−1)

and arg(ψηζ))) >arg ζ;

disjoint classes number turns to be ﬁnite

due to σKl>0, thus η= ord Klis ﬁnite for

any point ξ∈U(ul), and then

def

== K(1)

l∪K(2)

l∪. . . ∪K(η)

and along with that

#U(ul)def

== #K(1)

l+ #K(2)

l+. . . + #K(η)

3◦the colouring points from the set

UK(ul)def

== U(ul)∪ψ1(U(ul))∪. . .∪ψK(U(ul))

is separated on the ﬁnite disjoint classes

number (5) as well as U(ul) and full invari-

ant region colouring has been deﬁned by the

number #UK(ul).

Suppose ξl∈Υ be a accessible point from the

region Gland O(ξl)⊂Υ be an its trajectory. All

trajectory points ψk(ξl) are supplied with a self

neighbourhoods Uε(ψk(ξl)) of the same diameter

dεfor all k∈Z. Due to the compactness Υ there

subsists number N(ε), such that

Υ⊂[

µ6N(ε)

Uε(ψkµ(ξl)) def

== Ωε(Υ).

Then for any ε > 0 there exists k > K, such that

Ωε(Υ) ∩UK(ul)6= Ø, or that too

#(Ωε(Υ) ∩UK(ul)) >0.

Indeed for any colouring point a form of the map (1)

at iterations due to the boundary Υ stability turns

out to be suﬃciently close to Υ. Inter alia index

lfor the coverage Ωε(Υ) building can choose any

from 0, ν. Then any neighbourhood Uε(ψkµ(ξl))

contains all colours points. In addition the cov-

erage Ωε(Υ) turns out to be minimal in the sense

that

Ωε(Υ) \Uε(ψkµ(ξl))

is not connected for all µ∈1, N(ε).

All colours points full number in any neigh-

bourhood Uε(ψkµ(ξl)) by the number every colour-

ing points summation has been determined

#(Uε(ψkµ(ξl)) ∩G0) + . . . + #(Uε(ψkµ(ξl)) ∩Gν).

Theorem 3 Suppose Ki⊂UK(ui)⊂Giand

Kj⊂UK(uj)⊂Gj,i, j ∈0, ν, are colouring

points from the diﬀerent regions. Then

#U(ui): #U(uj) = σKi:σKj

for all ε > 0and suﬃciently large Kif

#(Uε(ψkµ(ξl)) ∩Ki): #(Uε(ψkµ(ξl)) ∩Kj)=1

for every µ∈1, N(ε).

An explanation of the term ¡¡for suﬃciently

large K¿¿ here and further will appear in the

course of the future narrative (see next section).

PROOF. If σKi=σKjthen the statement is

trivial. Therefore it is quite natural to assume

σKi6=σKjfor some i, j ∈0, ν and one can sup-

pose σKi> σKj. Then for every Kthe inequality

#K(1)

i>#K(1)

jfor UK(ui) and UK(uj) is faithful

in the event that #U(ui) = #U(uj). Thus

#(K(1)

i∪K(2)

i)>#(K(1)

j∪K(2)

for UK(ui) and UK(uj), because

K(1)

l∩K(2)

l= Ø

for all l∈0, ν, due to

σ(Ki⊕Ki)> σ(Kj⊕Kj).

Now let us increase the colouring points number

in neighbourhood U(uj) as follows, so that the

colouring points number of class K(1)

jhas been

increased to K(1)

jso that

#K(1)

i= #K(1)

j=α·#K(1)

thereby lengthening the sequence of indices of the

coloured points in the region Gj. Then for suf-

ﬁciently large K, the intersections Ωε(Υ) ∩K(1)

and Ωε(Υ) ∩K(1)

jpossess same density for any

ε > 0, i. e.

#(Ωε(Υ) ∩K(1)

i) = #(Ωε(Υ) ∩K(1)

j),

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so that

#(Ωε(Υ) ∩K(1)

i) = α·#(Ωε(Υ) ∩K(1)

j).

It is means that if αdef

== σKj:σKithen the colour-

ing point sets possess the same density for suﬃ-

ciently large K

σKj·#(Ωε(Υ) ∩K(1)

i) = σKi·#(Ωε(Υ) ∩K(1)

j).

on following condition

#(Uε(ψkµ(ξl)) ∩Ki): #(Uε(ψkµ(ξl)) ∩Kj)=1

is faithful for all ε > 0, if #U(ui): #U(uj) =

σKi:σKjut

Suppose for every region its colour has been

deﬁned in PostScript codes, for instance, in case

of two Wada basins and Wada ocean (ν= 2) as

follows 0 1 0 setrgbcolor

1 0 0 setrgbcolor

0 1 0 setrgbcolor

≈111

,(6)

or that too, the boundary is coloured to be white

colour. The formula (6) is faithful if N0:N1:N2≈

σK0:σK1:σK2due to ¡¡diﬀerent density¿¿ every

dense trajectories on the boundary.

Now suppose the positive semi-trajectories

O+(ζl), l ∈0, ν.

are colouring in self colours. Therefore all points

of O+(ζl) form ﬁnite disjoint classes number as

well as in 2◦, where Kl

def

== O+(ζl) and Klbe a

number of consecutive colouring points of O+(ζl)

(notation Kl

def

== #O+(ζl)). Then

#O+(ζl)def

== #K(1)

l+ #K(2)

l+. . . + #K(η)

for any ε > 0 there exists k > Kl, such that

Ωε(Υ) ∩O+(ζl)6= Ø, or that too

#(Ωε(Υ) ∩O+(ζl)) >0.

All colouring points full number in any neigh-

bourhood U(ψkµ(ξl)) from the coverage Ωε(Υ)

by the number every colouring points summation

has been determined

06l6ν

#(U(ψk(ξl)) ∩O+(ζl)).

Theorem 4 Suppose

Ki⊂O+(ζi)and Kj⊂O+(ζj),

i, j ∈0, ν, turn out to be colouring points from

the diﬀerent regions, such that O+(ζi)and O+(ζj)

are any two trajectories belonging to invariant re-

gions Giand Gjrespectively. Then

#O+(ζi): #O+(ζj) = σKi:σKj

for all ε > 0for suﬃciently long O+(ζi)and

O+(ζj)if

#(U(ψkµ(ξl)) ∩O+(ζi))

#(U(ψkµ(ξl)) ∩O+(ζj)) = 1

for every µ∈1, N(ε).

PROOF almost word for word repeats the pre-

vious theorem proof, but there are some nuances.

If σKi=σKjthen the statement is trivial. Then

one can suppose σKi> σKjfor some i, j ∈0, ν.

Therefore for every Kthe inequality #K(1)

#K(1)

jfor O+(ζi) and O+(ζj) is faithful in the

event that #O+(ζi) = #O+(ζj). Thus

#(K(1)

i∪K(2)

i)>#(K(1)

j∪K(2)

for O+(ζi) and O+(ζj), because

K(1)

l∩K(2)

l= Ø

for all l∈0, ν, due to

σ(Ki⊕Ki)> σ(Kj⊕Kj).

Let us increase the colouring points number in

trajectory O+(ζj) increasing the length of set the

colouring points, so that class K(1)

jhas been in-

creased to K(1)

jso, that

#K(1)

i= #K(1)

j=α·#K(1)

thereby lengthening the sequence of indices of the

coloured points in the region Gj. Then exactly

repeating the proof of the previous theorem one

complete the proof ut

Remark 4 The Schnirelmann densities, or that

too, rotation numbers for two diﬀer Wada basins

◦

Giand

◦

Gjdeﬁned by the formula (3) clearly in-

dicate on the Birrkhoﬀ curve ergodic properties.

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Indeed, densities for two colouring point sets of

the semi-trajectories O+(ζi)and O+(ζj)diﬀerent

length deﬁned by the expressions

inf

K6Ki

#(1, K ∩Ki)

Kand inf

K6Kj

#(1, K ∩Kj)

and their equality for σKi> σKj. Then subsists

suﬃciently large Ki= inf{Ki, Kj}, such that

inf

K6Ki

#(1, K ∩Ki)

K=α·inf

K6Ki

#(1, K ∩Kj)

or that too, α=σKi:σKj.

On other hand, let us place only point dif-

ferent from ﬁxed point in every invariant region

by colouring the trajectories points O(x0), O(x1)

and O(x2). Then the formula (6) is faithful if

k0:k1:k2≈σK0:σK1:σK2for the same reason.

Now let us formulate ¡¡synthetic condition¿¿,

exactly the condition connecting of the colouring

points number and the colouring semi-trajectories

length. In order to formulate the condition, one

suppose #O+(ζl) is the colouring semi-trajectories

length and #ζlis the number of colouring points,

such that for any pair colouring points #ζland

#ζ0

lthe intersection their semi-trajectories

O+(ζl)∩O+(ζ0

turn out to be empty. Then full colouring points

set has been deﬁned by the formula

def

== [

µ6#ζl

O+(ζµ

l) (7)

forming ﬁnite disjoint classes number as well as

in 2◦. Therefore full colouring points number has

been deﬁned by the formula

#Kl

def

== #O+(ζl)·#ζl,

so that Ωε(Υ) ∩Kl6= Ø, or that too

#(Ωε(Υ) ∩Kl)>0.

Theorem 5 Suppose Ki⊂Giand Kj⊂Gj,

i, j ∈0, ν (deﬁned by equality (7)), are colouring

points from the diﬀerent regions. Then

#O+(ζi)·#ζi

#O+(ζj)·#ζj

=σKi:σKj(8)

for all ε > 0for suﬃciently long O+(ζi)and

O+(ζj), and suﬃciently large Kiand Kjif

#(U(ψkµ(ξl)) ∩Ki): #(U(ψkµ(ξl)) ∩Kj)=1

for every µ∈1, N(ε).

This statement turns out to be synthetic. In-

deed, supposing

#O+(ζi)=#O+(ζj) or #ζi= #ζj

one comes to the conditions of previous theorems.

PROOF. In accordance with the remark 4 and

theorem 3 due to combination theorems 3 and 4

the formula (8) is obtained ut

Let us consider the following frequently oc-

curring dynamic situation, such that point p0is

the ﬁxed unstable antisaddle and everyone else

unstable antisaddles pl,l∈1, ν are ν-periodic

points.

Proposition 4 det #O+(ζi) #O+(ζj)

#ζj#ζi= 1

supposing #ζj6= #ζior #O+(ζi)6= #O+(ζj)for

all i, j ∈1, ν.

PROOF. This proposition turns out to be the

theorem 5 direct corollary. Indeed, if unstable

antisaddles piand pjare periodic then σKi=

σKjfor Wada basins ◦

Giand ◦

Gjrespectively, or

that too, there exist integers forming arithmetic

progression Aarbitrary long, such that

◦

Gi=ψa(◦

Gj)

for every a∈Aut

Let us call the 3-separatrix ﬁxed or periodic point

B-saddle, by the bike saddle image.

Birkhoﬀ curve containing the only B-saddle

turns out to be four regions common boundary.

One can prime example with rotary symmetry,

due to it three Wada basins have identical the

rotation number, in contrast to Wada ocean.

In paper [16], the authors did not solve the

regions colouring problem, because ones solved

5. B-saddle Colouring Example

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other problems. Inter alia the possibility of con-

structing the Y-connection volt-ampere charac-

teristic as the dynamic system action result has

been studied with necessary properties.

Omitting the preliminary justiﬁcations and

constructions (about this, see [16]) let us deﬁne

the dynamical system by the diﬀeomorphism ac-

tion on the plane at iterations as follows:

1◦at ﬁrst, one introduce the notations

˜xdef

== tanh(%cos φ) and ˜ydef

== tanh(%sin φ);

2◦Dynamic system deﬁned by action of map

in polar coordinates

Z(%, φ)7→ %·eiφ,(9)

where R=±1, T=±1 and

Z(%, φ)def

== TR˜x+ ˜y

p+iq˜yeiRϕ(˜x,˜y);

3◦then for the mapping from the formula (9),

the formula (by the simplest means) is con-

structed when moving in a positive direc-

tion

(|Z(%, φ)|,arg Z+ Φ(φ) + ∆) 7→ (%, φ) (10)

and when moving in a negative direction

(|Z(%, φ)|,arg Z+Φ(φ)−∆) 7→ (%, φ); (11)

where Φ(φ)def

== φ−φ(mod 2π);

4◦the formula for component |Z(%, φ)|is con-

structed as follows

||||Z(%, φ)|def

== sR˜x+ ˜y

p2

+q2y27→ %, (12)

while components arg Z, if R=1, and arg Z,

if R=−1, are deﬁned by the formulae

arg Zdef

== Rϕ(˜x, ˜y) + arctan qp˜y

R˜x+ ˜y,(13)

arg Zdef

== Rϕ(˜x, ˜y)−arctan qp˜y

R˜x+ ˜y; (14)

5◦let us make a natural replacement of vari-

ables in the formulae (10) and (11)

%·exp i(φ/Π−φ0)7→ u+iv,

where 2 ·Π∈Z\{0},φ0∈R;

Then the maps acting at iterations are de-

ﬁned by the formulae when moving in the

positive direction

|Z(%, φ)|,arg Z+ Φ(φ)+∆

Π7→ u+iv,

(15)

and when moving in the negative direction

|Z(%, φ)|,arg Z+ Φ(φ)−∆

Π7→ u+iv;

(16)

6◦therefore the replacement (u, v)7→ (%, φ)

occurs according to the formulae

pu2+v27→ %, Π·(φ0+arctan(v/u)+mπ)7→ φ,

for all m∈Z;

7◦then the formulae (15) and (16) are rewrit-

ten in the following form

|Z(%, φ)|,arg Z+ Γ(u, v)+∆

Π7→ u+iv, (17)

|Z(%, φ)|,arg Z+ Γ(u.v)−∆

Π7→ u+iv, (18)

where

Γ(u, v)def

== Φ(Π·(φ0+arctan(v/u)))+2πΠbm/2c;

8◦the variables ˜xand ˜yare written as follows

˜x= tan(√u2+v2cos(Π ·ω(u, v))),

˜y= tan(√u2+v2sin(Π ·ω(u, v))),

where ω(u, v)def

== φ0+ arctan(v/u) + mπ.

Now suppose R= 1 and Π = 3/2.

Remark 5 (on the technical details for the

colouring). Suppose for every invariant region

its colour has been deﬁned in PostScript codes,

for own instance, in case of three Wada basins

and Wada ocean (ν= 3) as follows

K0r0g0b0setrgbcolor

K1r1g1b1setrgbcolor

K2r2g2b2setrgbcolor

K3r3g3b3setrgbcolor

K1 1 1 (theorems condition),

(19)

or that too the boundary is summary coloured to

be white colour.

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DOI: 10.37394/232021.2024.4.5

Praskwya D. Serowa, Dmitry W. Serow

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Volume 4, 2024

Now one can colour the regions. There exist

four colouring variants accurate to the angle of

rotation.

Now back to the term ¡¡suﬃciently large K¿¿.

Suppose σK1=. . . =σKν> σK0then

Theorem 6 Suppose the rotations numbers such,

that σKl> σK0for colouring invariant regions G0

and Glrespectively. Then there subsists the se-

quence F0lfor every l∈1, ν, such that σF0l=

σK0:σKl.

PROOF. Suppose K(0)

0⊂G0and K(0)

l⊂Gl,

for all l∈1, ν (deﬁned by (7)), are colouring

points sets from two diﬀerent regions, such that

there exists ε0>0 for coverage Ωε0(Υ) and

#(U(ψkµ(ξl)) ∩K(0)

0): #(U(ψkµ(ξl)) ∩K(0)

l) = 1.

Now let us will increase consistently the colouring

points number in G0as follows

#K(1)

0=2#K(0)

0, . . . , #K(K)

0=(K+ 1)#K(0)

0, . . .

being an increasing arithmetic progression. The-

refore there exists decreasing sequence

ε1, ε2, . . . , εK, . . . ,

such that the following equalities

#(U(ψkµ(ξl)) ∩K(K)

l)= 1.(20)

are faithful for all K∈N. Then increasing se-

quence

#K(0)

l,#K(1)

l, . . . , #K(K)

l, . . .

in combination with own majoritarian increasing

sequence

#K(0)

l· 1,&#K(1)

#K(0)

l', . . . , &#K(K)

#K(0)

l', . . .!.

deliver the sequence

F0l: 1,&#K(1)

#K(0)

l', . . . , &#K(K)

#K(0)

l', . . . (21)

Thus from condition (20) in combination with the

theorem 5 the result is σF0l=σK0:σKlut

Corollary 2 If σKl=σK0then F0l≡N.

PROOF. Indeed, if σKl=σK0then

#K(K)

0= #K(K)

for all K∈N. Therefore F0l≡Nand σF0l≡1ut

Corollary 3 For any Birkhoﬀ curve Υ, there sub-

sists at least one sequence F0lsuch, that

0< σF0l<1.

However into practice far away not always

one can to deﬁnitely assert either σKl> σK0or

σKl< σK0, and even one can not to deﬁne either

σKl6=σK0or σKl=σK0. Nevertheless in such

an uncertain situation, for any pair of regions Gi

and Gjthe desire to ﬁnd a sequence of type Fij

remains relevant regardless of whether σKl6=σK0

or σKl=σK0.

Theorem 7 For any pair of invariant regions Gi

and Gjthere exists increasing sequence Fij , such

that

σFij =σKi:σKj

for all i, j ∈0, ν, if the following equality

#(U(ψkµ(ξl)) ∩K(K)

j)= 1.(22)

is faithful. Moreover increasing sequence Fij does

not depend from the construction method.

PROOF. The cases of σKi> σKjand σKi=

σKjhave been considered in the proofs of the

theorem 6 and corollary 2. Now suppose

#K(1)

j= 2#K(0)

j, . . . , #K(K)

j= (K+1)#K(0)

j, . . .

be an increasing arithmetic progression. Then

there exists decreasing sequence

ε1, ε2, . . . , εK, . . . ,

such that equality (22) is faithful for all K∈N.

Then every element of the arithmetic progression

corresponds to an element of the increasing se-

quence

#K(0)

i,#K(1)

i, . . . , #K(K)

i, . . .

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Praskwya D. Serowa, Dmitry W. Serow

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Volume 4, 2024

(a) (b)

Figure 1: Invariant colourings for three Wada basins and an Wada ocean with a common boundary

being Birkhoﬀ curve having the only ﬁxed point being inverse B- saddle for the dissipative action

ψat Π = 3/2 and R= 1, deﬁned by formula (17), in relation 2 : 4 : 4 : 5.

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Volume 4, 2024

The source of the ergodic theory for Wada basins

served from a remark in recent article [4]. Pro-

ceed on empirical and intuitive considerations,

the author made estimates of the relations of the

colouring densities of invariant regions, in order

to their common border turns to be discoloured

(i.e. white). The problem solution of the ¡¡bound-

ary discolouration¿¿ is turned out to be directly

related to the Wada basins ergodic properties.

The circle diﬀeomorphisms with irrational ro-

tation number torn out to be in a certain sense

simple rotations, or more exactly.

[1] Osipov, A. V. & Serow, D. W. Rotation

Number Additive Theory for Birkhoﬀ

Curves, NPCS, 20, 382 −393 (2017).

[2] Osipov, A. V. & Serow, D. W. Fractional

Densities for the Wada Basins, NPCS, 21,

389 −394 (2018).

[3] Osipov, A. V., Kovalew, I. A. & Serow, D. W.

Additive Dimension Theory for Birkhoﬀ

Curves, NPCS, 22, 164 −176 (2019).

[4] Serow, D. W. Nonwandering Continuum

Possessing Wada Property, Theor. and

Math. Phys., 207:3, 505 −520 (2021).

[5] Serow D. W. Dissipative Dynamical System

with Lakes of Vada, NPCS, 9, 4, 394 −98

(2006).

[6] Makarova M. V., Kovalew I. A. &

Serow D. W. Antisymmetric Wada Basins

Prime Example: Unstable Antisaddles Case,

NPCS, 21, 2, 188 −193 (2018).

[7] Furstenberg, H. Recurrence in Ergodic The-

ory and Combinatorial Number Theory,

Princeton University Press, Princeton, NJ,

1981.

[8] Khinchin A. Ya. Three Pearls of Number

Theory, Mineola, NY: Dover, 1998.

[9] Kennedy J., Yorke J. A. Basins of Wada,

Physica D 51: 213 −225 (1991).

[10] Kennedy J. ”A brief history of indecompos-

able continua” in Continua, edited by H.

Cook, W. T. Ingram, K. T. Kuperberg, A.

Lelek, and P. Minc (Marcel Dekker, New

York,1995), 103 −126.

[11] Coudene Y. Pictures of Hyperbolic Dynami-

cal Systems. NOTICES of the AMS, 53, 1,

8−13 (2006).

[12] Kuratowski, K. Topology, vol. I & vol. II,

Academic Press, New York & London, 1966.

[13] Borsuk K. Theory of Retracts. PWN,

Warszawa, 1967.

[14] Sierpinski, W. General Topology, University

of Toronto Press, Toronto (1952).

[15] Birkhoﬀ G. D. Sur quelques courbes ferm´es

remarquables, Bull. Soc. Math. France 60,

1−26 (1932).

[16] Maygula N. V., Serowa P. D., Kovalew I. A.,

Serow D. W., Plane Quazicrystals and Four

Colours Problem: Trivalent Graph Element

Construction, Nonlinear Dynamics and Ap-

plications, 26, 346 −357 (2020).

[17] Kovalew I. A. & Serow D. W. Illustrations of

Irreducibility and Tops of Umbrellas in the

PostScript Methodology, NPCS, 17, 318 −

326 (2014).

6. Conclusion

References

Contribution of Individual Authors to the

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research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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(Attribution 4.0 International, CC BY 4.0)

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_US

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DOI: 10.37394/232021.2024.4.5

Praskwya D. Serowa, Dmitry W. Serow

E-ISSN: 2732-9976

Volume 4, 2024