aarjaneiro/planetmath huggingface dataset:
highly-complex system diagrams

highly-complex system diagrams

0.1

Highly-complex systems

0.1.1

Introduction

Modeling the emergence of the

ultra-complex system

of the
human mind–based on the super-complex human organism– one needs to consider an associated
progression towards

higher dimensional algebras

from the lower dimensions of human neural network dynamics and the simple algebra of physical dynamics, as shown in the following, essentially

non-commutative

categorical

diagram

.
One can

represent

by square categorical diagrams the emergence of ultra-complex
dynamics from the super-complex dynamics of human organisms coupled

via

social interactions
in

characteristic

patterns represented by Rosetta biogroupoids (

http://planetmath.org/RosettaGroupoids

), together with the complex–albeit inanimate–systems with ‘chaos’ as discussed next.

0.2

Diagrams of simple and highly-complex systems

Definition 0.1

.

An

ultra-complex system,

U

C

⁢

S

is defined as an object

representation

in the following non-commutative
diagram of systems and

dynamic system

morphisms

or ‘dynamic

transformations

’:

\xymatrix

@

C

=

5

p

c

[

S

U

P

E

R

-

C

O

M

P

L

E

X

]

\ar

[

r

]

(

Higher
Dim

)

\ar

[

d

]

Λ

&

(

U

C

⁢

S

=

U

L

T

R

A

-

C

O

M

P

L

E

X

)

\ar

[

d

]

o

⁢

n

⁢

t

⁢

o

C

O

M

P

L

E

X

&

\ar

[

l

]

(

Generic Map

)

[

S

I

M

P

L

E

]

0.3

Remarks

Note that the above diagram is indeed not ‘natural’ (i.e. it is not

commutative

) for reasons
related to the emergence of the higher dimensions of the super–complex
(biological/organismic) and/or ultra–complex (psychological/neural network dynamic) levels in comparison with
the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.

An

ultra-complex system

represents the human mind and consciousness from the standpoint
of a categorical ontology theory of levels as the highest level of

complexity

that
emerged through biological and social coevolution over the last

2.2

million years on Earth.

Title

highly-complex system diagrams

Canonical name

HighlycomplexSystemDiagrams

Date of creation

2013-03-22 18:30:13

Last modified on

2013-03-22 18:30:13

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

7

Author

bci1 (20947)

Entry type

Topic

Classification

msc 37B25

Classification

msc 93D15

Classification

msc 18A05

Classification

msc 93D21

Classification

msc 37B10

Related topic

RosettaGroupoids

Related topic

StrongAIThesis

Related topic

CategoricalDynamics

Generated on Fri Feb 9 20:45:49 2018 by

LaTeXML
locally maximal

locally maximal

Let

f

:

M

→

M

be a

diffeomorphism

of a

smooth manifold

M

, and let

Λ

be a

hyperbolic set

for

f

. We say that

Λ

is

locally maximal

or

basic

if there exists a

neighborhood

U

of

Λ

such that

Λ

=

⋂

n

=

-

∞

∞

f

n

⁢

(

U

¯

)

.

Title

locally maximal

Canonical

name

LocallyMaximal

Date of creation

2013-03-22 14:07:24

Last modified on

2013-03-22 14:07:24

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37B35

Classification

msc 37D05

Synonym

basic set

Generated on Fri Feb 9 20:45:50 2018 by

LaTeXML
topological entropy

topological entropy

Let

(

X

,

d

)

be a

compact

metric space

and

f

:

X

→

X

a

continuous map

.
For each

n

≥

0

, we define a new metric

d

n

by

d

n

⁢

(

x

,

y

)

=

max

⁡

{

d

⁢

(

f

i

⁢

(

x

)

,

f

i

⁢

(

y

)

)

:

0

≤

i

<

n

}

.

Two points are

ϵ

-close with respect to this metric if their first

n

iterates

are

ϵ

-close.
For

ϵ

>

0

and

n

≥

0

we say that

F

⊂

X

is an

(

n

,

ϵ

)

-separated set if for each pair

x

,

y

of points of

F

we have

d

n

⁢

(

x

,

y

)

>

ϵ

. Denote by

N

⁢

(

n

,

ϵ

)

the

maximum

cardinality

of an

(

n

,

ϵ

)

-separated set (which is finite, because

X

is compact). Roughly,

N

⁢

(

n

,

ϵ

)

represents the number of “distinguishable” orbit

segments

of length

n

, assuming we cannot distinguish points that are less than

ϵ

apart.
The topological entropy of

f

is defined by

h

t

⁢

o

⁢

p

⁢

(

f

)

=

lim

ϵ

→

0

⁡

(

lim sup

n

→

∞

⁡

1

n

⁢

log

⁡

N

⁢

(

n

,

ϵ

)

)

.

It is easy to see that this limit always exists, but it could be

infinite

.
A rough

interpretation

of this number is that it measures the

average

exponential growth of the number of distinguishable orbit segments. Hence, roughly speaking again, we could say that the higher the topological entropy is, the more essentially different orbits we have.

Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew, with a different (but

equivalent

) definition to the one presented here. The definition we give here is due to Bowen and Dinaburg.

Title

topological entropy

Canonical name

TopologicalEntropy

Date of creation

2013-03-22 14:31:34

Last modified on

2013-03-22 14:31:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 37B40

Synonym

entropy

Generated on Fri Feb 9 20:45:53 2018 by

LaTeXML
expansive

expansive

If

(

X

,

d

)

is a

metric space

, a

homeomorphism

f

:

X

→

X

is said to be

expansive

if there is a

constant

ε

0

>

0

, called the

expansivity constant

, such that for any two points of

X

, their

n

-th iterates are at least

ε

0

apart for some integer

n

; i.e. if for any pair of points

x

≠

y

in

X

there is

n

∈

ℤ

such that

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

≥

ε

0

.

The space

X

is often assumed to be

compact

, since under that

assumption

expansivity is a topological property; i.e. any map which is

topologically conjugate

to

f

is expansive if

f

is expansive (possibly with a different expansivity constant).

If

f

:

X

→

X

is a

continuous map

, we say that

X

is

positively expansive

(or forward expansive) if there is

ε

0

such that, for any

x

≠

y

in

X

, there is

n

∈

ℕ

such that

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

≥

ε

0

.

Remarks.

The latter condition is much

stronger

than expansivity. In fact, one can prove that if

X

is compact and

f

is a positively expansive homeomorphism, then

X

is finite (proof (

http://planetmath.org/OnlyCompactMetricSpacesThatAdmitAPostivelyExpansiveHomeomorphismAreDiscreteSpaces

)).

Title

expansive

Canonical name

Expansive

Date of creation

2013-03-22 13:47:48

Last modified on

2013-03-22 13:47:48

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

14

Author

Koro (127)

Entry type

Definition

Classification

msc 37B99

Defines

expansivity

Defines

positively expansive

Defines

forward expansive

Generated on Fri Feb 9 20:45:55 2018 by

LaTeXML
omega limit set

omega limit set

Let

Φ

⁢

(

t

,

x

)

be the flow of the

differential equation

x

′

=

f

⁢

(

x

)

, where

f

∈

C

k

⁢

(

M

,

ℝ

n

)

, with

k

≥

1

and

M

an open subset of

ℝ

n

.
Consider

x

∈

M

.

The

omega limit set

of

x

, denoted

ω

⁢

(

x

)

, is the set of points

y

∈

M

such that there exists a sequence

t

n

→

∞

with

Φ

⁢

(

t

n

,

x

)

=

y

.

Similarly, the alpha limit set of

x

, denoted

α

⁢

(

x

)

, is the set of points

y

∈

M

such that there exists a sequence

t

n

→

-

∞

with

Φ

⁢

(

t

n

,

x

)

=

y

.

Note that the definition is the same for more general

dynamical systems

.

Title

omega limit set

Canonical name

OmegaLimitSet

Date of creation

2013-03-22 13:18:42

Last modified on

2013-03-22 13:18:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 37B99

Classification

msc 34C05

Synonym

ω

-limit set

Synonym

α

-limit set

Related topic

LimitCycle

Defines

alpha limit set

Generated on Fri Feb 9 20:45:57 2018 by

LaTeXML
topologically transitive

topologically transitive

A

continuous

surjection

f

on a

topological space

X

to itself
is

topologically transitive

if for every
pair of

open sets

U

and

V

in

X

there is an integer

n

>

0

such that

f

n

⁢

(

U

)

∩

V

≠

∅

, where

f

n

denotes the

n

-th iterate of

f

.

If for every pair of open sets

U

and

V

there is an integer

N

such that

f

n

⁢

(

U

)

∩

V

≠

∅

for each

n

>

N

, we say that

f

is

topologically mixing

.

If

X

is a

compact

metric space

, then

f

is topologically transitive if and only if there exists a point

x

∈

X

with a dense orbit, i.e. such that

𝒪

⁢

(

x

,

f

)

=

{

f

n

⁢

(

x

)

:

n

∈

ℕ

}

is dense in

X

.

Title

topologically transitive

Canonical name

TopologicallyTransitive

Date of creation

2013-03-22 13:41:05

Last modified on

2013-03-22 13:41:05

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37B99

Classification

msc 54H20

Defines

topologically mixing

Defines

topological mixing

Generated on Fri Feb 9 20:46:02 2018 by

LaTeXML
uniform expansivity

uniform expansivity

Let

(

X

,

d

)

be a

compact

metric space

and let

f

:

X

→

X

be an

expansive

homeomorphism

.

Theorem (

uniform expansivity

).

For every

ϵ

>

0

and

δ

>

0

there is

N

>

0

such that for each pair

x

,

y

of points of

X

such that

d

⁢

(

x

,

y

)

>

ϵ

there is

n

∈

ℤ

with

|

n

|

≤

N

such that

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

>

c

-

δ

, where

c

is the expansivity constant of

f

.

Proof.

Let

K

=

{

(

x

,

y

)

∈

X

×

X

:

d

⁢

(

x

,

y

)

≥

ϵ

/

2

}

. Then K is closed, and hence compact. For each pair

(

x

,

y

)

∈

K

, there is

n

(

x

,

y

)

∈

ℤ

such that

d

⁢

(

f

n

(

x

,

y

)

⁢

(

x

)

,

f

n

(

x

,

y

)

⁢

(

y

)

)

≥

c

. Since the mapping

F

:

X

×

X

→

X

×

X

defined by

F

⁢

(

x

,

y

)

=

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

is

continuous

,

F

n

x

is also continuous and there is a

neighborhood

U

(

x

,

y

)

of each

(

x

,

y

)

∈

K

such that

d

⁢

(

f

n

(

x

,

y

)

⁢

(

u

)

,

f

n

(

x

,

y

)

⁢

(

v

)

)

<

c

-

δ

for each

(

u

,

v

)

∈

U

(

x

,

y

)

. Since

K

is compact and

{

U

(

x

,

y

)

:

(

x

,

y

)

∈

K

}

is an

open cover

of

K

, there is a finite subcover

{

U

(

x

i

,

y

i

)

:

1

≤

i

≤

m

}

. Let

N

=

max

⁡

{

|

n

(

x

i

,

y

i

)

|

:

1

≤

i

≤

m

}

. If

d

⁢

(

x

,

y

)

>

ϵ

, then

(

x

,

y

)

∈

K

, so that

(

x

,

y

)

∈

U

(

x

i

,

y

i

)

for some

i

∈

{

1

,

…

,

m

}

. Thus for

n

=

n

(

x

i

,

y

i

)

we have

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

<

c

-

δ

and

|

n

|

≤

N

as requred.

Title

uniform expansivity

Canonical name

UniformExpansivity

Date of creation

2013-03-22 13:55:15

Last modified on

2013-03-22 13:55:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 37B99

Generated on Fri Feb 9 20:46:05 2018 by

LaTeXML
ω-limit set

ω

-limit set

Let

X

be a

metric space

, and let

f

:

X

→

X

be a

homeomorphism

.
The

ω

-limit set

of

x

∈

X

, denoted by

ω

⁢

(

x

,

f

)

, is the set of

cluster points

of the forward orbit

{

f

n

⁢

(

x

)

}

n

∈

ℕ

.
Hence,

y

∈

ω

⁢

(

x

,

f

)

if and only if there is a

strictly increasing

sequence

of

natural numbers

{

n

k

}

k

∈

ℕ

such that

f

n

k

⁢

(

x

)

→

y

as

k

→

∞

.

Another way to express this is

ω

⁢

(

x

,

f

)

=

⋂

n

∈

ℕ

{

f

k

⁢

(

x

)

:

k

>

n

}

¯

.

The

α

-limit set

is defined in a

similar

fashion, but for the backward orbit; i.e.

α

⁢

(

x

,

f

)

=

ω

⁢

(

x

,

f

-

1

)

.

Both sets are

f

-

invariant

, and if

X

is

compact

, they are compact and nonempty.

If

φ

:

ℝ

×

X

→

X

is a continuous flow, the

definition

is similar:

ω

⁢

(

x

,

φ

)

consists of those

elements

y

of

X

for which there exists a strictly increasing sequnece

{

t

n

}

of

real numbers

such that

t

n

→

∞

and

φ

⁢

(

x

,

t

n

)

→

y

as

n

→

∞

.
Similarly,

α

⁢

(

x

,

φ

)

is the

ω

-limit set of the reversed flow (i.e.

ψ

⁢

(

x

,

t

)

=

ϕ

⁢

(

x

,

-

t

)

).
Again, these sets are invariant and if

X

is compact they are compact and nonempty. Furthermore,

ω

⁢

(

x

,

f

)

=

⋂

n

∈

ℕ

{

φ

⁢

(

x

,

t

)

:

t

>

n

}

¯

.

Title

ω

-limit set

Canonical

name

omegalimitSet

Date of creation

2013-03-22 13:39:37

Last modified on

2013-03-22 13:39:37

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 37B99

Synonym

omega-limit set

Related topic

NonwanderingSet

Defines

α

-limit

Defines

alpha-limit

Defines

ω

-limit

Defines

omega-limit

Generated on Fri Feb 9 20:46:07 2018 by

LaTeXML
Hayashi’s connecting lemma

Hayashi’s connecting lemma

Let

f

:

M

→

M

be a

C

1

diffeomorphism

of the compact

smooth manifold

M

, and let

p

,

q

∈

M

be such that there exists a nonperiodic point in

ω

⁢

(

p

,

f

)

∩

α

⁢

(

q

,

f

)

(the intersection of the

alpha limit set

of

q

with the omega limit set of

p

). Then there exists a diffeomorphism

g

, arbitrarily close to

f

in the

𝒞

1

topology of

Diff

1

⁡

(

M

)

, such that

q

is in the forward orbit of

p

through

g

, i.e. such that

g

n

⁢

(

p

)

=

q

for some

n

>

0

.

References

1

Wen, L., Xia, Z.,

C

1

connecting lemmas

, Trans. Amer. Math. Soc.

352

(2000), no. 11, 5213-5230.

Title

Hayashi’s connecting lemma

Canonical name

HayashisConnectingLemma

Date of creation

2013-03-22 14:07:16

Last modified on

2013-03-22 14:07:16

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C05

Classification

msc 37C25

Synonym

connecting lemma

Generated on Fri Feb 9 20:46:09 2018 by

LaTeXML
flow

flow

A

flow

on a set

X

is a group action of

(

ℝ

,

+

)

on

X

.

More explicitly, a flow is a function

φ

:

X

×

ℝ

→

X

satisfying the following

properties

:

1.

φ

⁢

(

x

,

0

)

=

x

2.

φ

⁢

(

φ

⁢

(

x

,

t

)

,

s

)

=

φ

⁢

(

x

,

s

+

t

)

for all

s

,

t

in

ℝ

and

x

∈

X

.

The set

𝒪

⁢

(

x

,

φ

)

=

{

φ

⁢

(

x

,

t

)

:

t

∈

ℝ

}

is called the orbit of

x

by

φ

.

Flows are usually required to be

continuous

or

differentiable

, when the space

X

has some additional structure (e.g. when

X

is a

topological space

or when

X

=

ℝ

n

.)

The most common examples of flows arise from describing the solutions of the

autonomous

ordinary differential equation

y

′

=

f

⁢

(

y

)

,

y

⁢

(

0

)

=

x

(1)

as a function of the

initial condition

x

, when the equation has existence and uniqueness of solutions.
That is, if (

1

) has a unique solution

ψ

x

:

ℝ

→

X

for each

x

∈

X

, then

φ

⁢

(

x

,

t

)

=

ψ

x

⁢

(

t

)

defines a flow.

Title

flow

Canonical

name

Flow1

Date of creation

2013-03-22 13:12:34

Last modified on

2013-03-22 13:12:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 37C10

Generated on Fri Feb 9 20:46:10 2018 by

LaTeXML
Frobenius’ theorem

Frobenius’ theorem

Theorem

(Frobenius).

Let

M

be a

smooth manifold

(

C

∞

) and let

Δ

be a

distribution

on

M

. Then

Δ

is

completely integrable

if and only if

Δ

is involutive.

One

direction

in the proof is pretty easy since the

tangent space

of an integral manifold is involutive, so sometimes the theorem is only
stated in one direction, that is: If

Δ

is involutive then it is
completely integrable.

Another way to the theorem is that if we have

n

vector fields

{

X

k

}

k

=

1

n

on a manifold

M

such that they are

linearly independent

at every point of the manifold, and furthermore if for any

k

,

m

we have

[

X

k

,

X

m

]

=

∑

j

=

1

n

a

j

⁢

X

j

for some

C

∞

functions

a

j

, then for any point

x

∈

N

, there exists a germ of a

submanifold

N

⊂

M

, through

x

, such that

T

⁢

N

is spanned
by

{

X

k

}

k

=

1

n

. Note that if we extend

N

to all of

M

, it need not be
an embedded submanifold anymore, but just an immersed one.

For

n

=

1

above, this is just the

existence and uniqueness of solution of ordinary differential equations

.

References

1

William M. Boothby.
,
Academic Press, San Diego, California, 2003.

2

Frobenius theorem

at Wikipedia:

http://en.wikipedia.org/wiki/Frobenius_theorem

http://en.wikipedia.org/wiki/Frobenius_theorem

Title

Frobenius’ theorem

Canonical name

FrobeniusTheorem

Date of creation

2013-03-22 14:52:03

Last modified on

2013-03-22 14:52:03

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

9

Author

jirka (4157)

Entry type

Theorem

Classification

msc 37C10

Classification

msc 53-00

Classification

msc 53B25

Related topic

Distribution5

Related topic

IntegralManifold

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globally attracting fixed point

globally attracting fixed point

An

attracting fixed point

is considered globally attracting if its

stable manifold

is the entire space. Equivalently, the

fixed point

x

*

is globally attracting if for all

x

,

x

⁢

(

t

)

→

x

*

as

t

→

∞

.

Title

globally attracting fixed point

Canonical name

GloballyAttractingFixedPoint

Date of creation

2013-03-22 13:06:27

Last modified on

2013-03-22 13:06:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 37C10

Related topic

AttractingFixedPoint

Generated on Fri Feb 9 20:46:14 2018 by

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topological conjugation

topological conjugation

Let

X

and

Y

be

topological spaces

, and let

f

:

X

→

X

and

g

:

Y

→

Y

be

continuous functions

. We say that

f

is

topologically semiconjugate

to

g

, if there exists a

continuous

surjection

h

:

Y

→

X

such that

f

⁢

h

=

h

⁢

g

. If

h

is a

homeomorphism

,
then we say that

f

and

g

are

topologically conjugate

, and we call

h

a

topological conjugation

between

f

and

g

.

Similarly, a flow

φ

on

X

is topologically semiconjugate to a flow

ψ

on

Y

if there is a continuous surjection

h

:

Y

→

X

such that

φ

⁢

(

h

⁢

(

y

)

,

t

)

=

h

⁢

ψ

⁢

(

y

,

t

)

for each

y

∈

Y

,

t

∈

ℝ

. If

h

is a homeomorphism then

ψ

and

φ

are topologically conjugate.

0.1

Remarks

Topological conjugation defines an

equivalence relation

in the
space of all continuous surjections of a topological space to itself,
by declaring

f

and

g

to be related if they are topologically
conjugate. This equivalence relation is very useful in the theory of

dynamical systems

, since each class contains all

functions

which
share the same dynamics from the topological viewpoint. In fact, orbits
of

g

are mapped to homeomorphic orbits of

f

through the

conjugation

.
Writing

g

=

h

-

1

⁢

f

⁢

h

makes this fact evident:

g

n

=

h

-

1

⁢

f

n

⁢

h

.
Speaking informally, topological conjugation is a “change of coordinates” in the topological sense.

However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps

φ

⁢

(

⋅

,

t

)

and

ψ

⁢

(

⋅

,

t

)

to be topologically conjugate for each

t

, which is requiring more than simply that orbits of

φ

be mapped to orbits of

ψ

homeomorphically.
This motivates the definition of

topological equivalence

, which also

partitions

the set of all flows in

X

into classes of flows sharing the same dynamics, again from the topological viewpoint.

We say that

ψ

and

φ

are

topologically equivalent

, if there is an homeomorphism

h

:

Y

→

X

,

mapping

orbits of

ψ

to orbits of

φ

homeomorphically, and preserving orientation of the orbits. This means that:

1.

h

⁢

(

𝒪

⁢

(

y

,

ψ

)

)

=

{

h

⁢

(

ψ

⁢

(

y

,

t

)

)

:

t

∈

ℝ

}

=

{

φ

⁢

(

h

⁢

(

y

)

,

t

)

:

t

∈

ℝ

}

=

𝒪

⁢

(

h

⁢

(

y

)

,

φ

)

for each

y

∈

Y

;

2.

for each

y

∈

Y

, there is

δ

>

0

such that, if

0

<

|

s

|

<

t

<

δ

, and if

s

is such that

φ

⁢

(

h

⁢

(

y

)

,

s

)

=

h

⁢

(

ψ

⁢

(

y

,

t

)

)

, then

s

>

0

.

Title

topological conjugation

Canonical name

TopologicalConjugation

Date of creation

2013-03-22 13:41:02

Last modified on

2013-03-22 13:41:02

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

14

Author

Koro (127)

Entry type

Definition

Classification

msc 37C15

Classification

msc 37B99

Defines

topologically conjugate

Defines

topological semiconjugation

Defines

topologically semiconjugate

Defines

topologically equivalent

Defines

topological equivalence

Generated on Fri Feb 9 20:46:16 2018 by

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Kupka-Smale theorem

Kupka-Smale theorem

Let

M

be a compact

smooth manifold

, and let

Diff

k

⁡

(

M

)

denote the space of

𝒞

k

diffeomorphisms

from

M

to itself. For every

k

∈

ℕ

, the set of

Kupka-Smale

diffeomorphisms is residual (

http://planetmath.org/Residual

) in the

𝒞

k

Whitney topology

of

Diff

k

⁡

(

M

)

.

Title

Kupka-Smale theorem

Canonical name

KupkaSmaleTheorem

Date of creation

2013-03-22 13:41:16

Last modified on

2013-03-22 13:41:16

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C20

Related topic

KupkaSmale

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Pugh’s closing lemma

Pugh’s closing lemma

Let

f

:

M

→

M

be a

𝒞

1

diffeomorphism

of a compact

smooth manifold

M

. Given a

nonwandering point

x

of

f

, there exists a diffeomorphism

g

arbitrarily close to

f

in the

𝒞

1

topology of

Diff

1

⁡

(

M

)

such that

x

is a periodic point of

g

.

The analogous theorem holds when

x

is a nonwandering point of a

𝒞

1

flow on

M

.

References

1

Pugh, C.,

An improved

closing lemma

and a

general density theorem

, Amer. J. Math.

89

(1967).

Title

Pugh’s closing lemma

Canonical name

PughsClosingLemma

Date of creation

2013-03-22 14:07:13

Last modified on

2013-03-22 14:07:13

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C20

Classification

msc 37C25

Synonym

closing lemma

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Pugh’s general density theorem

Pugh’s general density theorem

Let

M

be a compact

smooth manifold

. There is a residual subset of

Diff

1

⁡

(

M

)

in which every element

f

satisfies

Per

⁡

(

f

)

¯

=

Ω

⁢

(

f

)

.
In other words: Generically, the set of periodic points of a

𝒞

1

diffeomorphism

is dense in its

nonwandering set

.

Here,

Diff

1

⁡

(

M

)

denotes the set of all

𝒞

1

difeomorphisms from

M

to
itself, endowed with the (strong)

𝒞

1

topology.

References

1

Pugh, C.,

An improved

closing lemma

and a

general density theorem

, Amer. J. Math.

89

(1967).

Title

Pugh’s general density theorem

Canonical name

PughsGeneralDensityTheorem

Date of creation

2013-03-22 13:40:42

Last modified on

2013-03-22 13:40:42

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C20

Classification

msc 37C25

Synonym

general density theorem

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