aarjaneiro/planetmath huggingface dataset:
structural stability

structural stability

Given a

metric space

(

X

,

d

)

and an

homeomorphism

f

:

X

→

X

, we say that

f

is

structurally stable

if there is a

neighborhood

𝒱

of

f

in

Homeo

⁡

(

X

)

(the space of all homeomorphisms mapping

X

to itself endowed with the

compact-open topology

) such that every element of

𝒱

is

topologically conjugate

to

f

.

If

M

is a

compact

smooth manifold

, a

𝒞

k

diffeomorphism

f

is said to be

𝒞

k

structurally stable if there is a neighborhood of

f

in

Diff

k

⁡

(

M

)

(the space of all

𝒞

k

diffeomorphisms from

M

to itself endowed with the strong

𝒞

k

topology

) in which every element is topologically conjugate to

f

.

If

X

is a vector field in the smooth manifold

M

, we say that

X

is

𝒞

k

-structurally stable if there is a neighborhood of

X

in

𝒳

k

⁢

(

M

)

(the space of all

𝒞

k

vector fields on

M

endowed with the strong

𝒞

k

topology) in which every element is topologically equivalent to

X

, i.e. such that every other field

Y

in that neighborhood generates a flow on

M

that is topologically equivalent to the flow generated by

X

.

Remark.

The concept of structural stability may be generalized to other

spaces of functions

with other topologies; the general idea is that a function or flow is structurally stable if any other function or flow close enough to it has similar dynamics (from the topological viewpoint), which essentially means that the dynamics will not change under

small perturbations

.

Title

structural stability

Canonical name

StructuralStability

Date of creation

2013-03-22 13:48:45

Last modified on

2013-03-22 13:48:45

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 37C20

Classification

msc 34D30

Synonym

structurally stable

Generated on Fri Feb 9 20:46:22 2018 by

LaTeXML
Hartman-Grobman theorem

Hartman-Grobman theorem

Let

U

and

V

be

open subsets

of a

Banach space

E

such that

0

∈

U

∩

V

.
If a

diffeomorphism

f

:

U

→

V

has

0

as a

hyperbolic fixed
point

, then

f

and

D

⁢

f

⁢

(

0

)

are locally

topologically conjugate

at

0

,
i.e. there are

neighborhoods

U

~

and

V

~

of

0

and a

homeomorphism

h

:

U

~

→

V

~

such that

D

⁢

f

⁢

(

0

)

⁢

h

=

h

⁢

f

.

Title

Hartman-Grobman theorem

Canonical name

HartmanGrobmanTheorem

Date of creation

2013-03-22 14:25:15

Last modified on

2013-03-22 14:25:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C25

Synonym

Grobman-Hartman theorem

Synonym

Hartman’s theorem

Generated on Fri Feb 9 20:46:24 2018 by

LaTeXML
hyperbolic fixed point

hyperbolic fixed point

Let

M

be a

smooth manifold

. A

fixed point

x

of a

diffeomorphism

f

:

M

→

M

is said to be a

hyperbolic fixed point

if

D

⁢

f

⁢

(

x

)

is a

linear hyperbolic isomorphism

. If

x

is a periodic point of least period

n

, it is called a

hyperbolic periodic point

if it is a hyperbolic fixed point of

f

n

(the

n

-th iterate of

f

).

If the

dimension

of the

stable manifold

of a fixed point is zero, the point is called a

source

; if the dimension of its unstable manifold is zero, it is called a

sink

; and if both the stable and unstable manifold have nonzero dimension, it is called a

saddle

.

Title

hyperbolic fixed point

Canonical name

HyperbolicFixedPoint

Date of creation

2013-03-22 13:47:57

Last modified on

2013-03-22 13:47:57

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 37C25

Classification

msc 37D05

Related topic

StableManifold

Related topic

HyperbolicSet

Defines

hyperbolic periodic point

Defines

source

Defines

sink

Defines

saddle

Generated on Fri Feb 9 20:46:25 2018 by

LaTeXML
heteroclinic

heteroclinic

Let

f

be an

homeomorphism

mapping a

topological space

X

to itself or a flow on

X

. An

heteroclinic point

, or

heteroclinic

intersection, is a point that belongs to the intersection of the

stable set

of

x

with the unstable set of

y

, where

x

and

y

are two different fixed or periodic points of

f

, i.e. a point that belongs to

W

s

⁢

(

f

,

x

)

∩

W

u

⁢

(

f

,

y

)

.

Title

heteroclinic

Canonical name

Heteroclinic

Date of creation

2013-03-22 13:48:38

Last modified on

2013-03-22 13:48:38

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 37C29

Generated on Fri Feb 9 20:46:34 2018 by

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homoclinic

homoclinic

If

X

is a

topological space

and

f

is a flow on

X

or an

homeomorphism

mapping

X

to itself, we say that

x

∈

X

is an

homoclinic point

(or

homoclinic

intersection) if it belongs to both the stable and

unstable sets

of some fixed or periodic point

p

; i.e.

x

∈

W

s

⁢

(

f

,

p

)

∩

W

u

⁢

(

f

,

p

)

.

The orbit of an homoclinic point is called an homoclinic orbit.

Title

homoclinic

Canonical name

Homoclinic

Date of creation

2013-03-22 13:48:35

Last modified on

2013-03-22 13:48:35

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37C29

Generated on Fri Feb 9 20:46:36 2018 by

LaTeXML
homoclinic class

homoclinic class

Let

M

be a

compact

smooth manifold

and

f

:

M

→

M

a

diffeomorphism

. The

homoclinic class

of a

hyperbolic periodic point

p

of

f

, denoted

H

⁢

(

p

,

f

)

, is the

closure

of the set of

transverse

intersections

between the stable and

unstable manifolds

all points in the orbit of

p

; i.e.

H

⁢

(

p

,

f

)

=

cl

⁡

(

⋃

n

∈

ℕ

W

s

⁢

(

p

)

⋔

⋃

n

∈

ℤ

W

u

⁢

(

p

)

)

.

Homoclinic classes are

topologically transitive

, and the number of homoclinic classes is at most

countable

. Moreover,

generically

(in the

𝒞

1

topology

of

Diff

⁡

(

M

)

), they are

pairwise disjoint

and maximally

transitive

.

Title

homoclinic class

Canonical name

HomoclinicClass

Date of creation

2013-03-22 14:07:30

Last modified on

2013-03-22 14:07:30

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 37C29

Generated on Fri Feb 9 20:46:38 2018 by

LaTeXML
pseudo-orbit

pseudo-orbit

Let

(

X

,

d

)

be a

metric space

,

f

:

X

→

X

a function, and let

ϵ

>

0

. An

ϵ

-pseudo-orbit for

f

is a sequence

{

x

n

:

n

∈

ℤ

,

a

<

n

<

b

}

, where

-

∞

≤

a

<

b

≤

∞

, such that

d

⁢

(

x

n

+

1

,

f

⁢

(

x

n

)

)

<

ϵ

for all

a

<

n

<

b

. A periodic pseudo-orbit is an infinite pseudo-orbit

{

x

n

}

such that there is some

p

with

x

n

+

p

=

x

n

for all n.

Given

δ

>

0

, the pseudo-orbit

{

x

n

:

a

<

n

<

b

}

is said to be

δ

-shadowed by the orbit of

x

, if

d

⁢

(

x

n

,

f

n

⁢

(

x

)

)

<

δ

for all

a

<

n

<

b

.

Title

pseudo-orbit

Canonical name

Pseudoorbit

Date of creation

2013-03-22 14:07:19

Last modified on

2013-03-22 14:07:19

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 37C50

Synonym

pseudo orbit

Synonym

ϵ

-pseudo-orbit

Synonym

ϵ

-orbit

Synonym

ϵ

-chain

Defines

shadow

Defines

shadowing

Defines

shadowed

Generated on Fri Feb 9 20:46:39 2018 by

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shadowing lemma

shadowing lemma

Let

M

be a

Riemannian manifold

,

f

:

M

→

M

a

diffeomorphism

and

Λ

⊂

M

a

compact

hyperbolic set

for

f

. Then there is a

neighborhood

U

of

Λ

such that for every

δ

>

0

there is an

ϵ

>

0

so that every

ϵ

-orbit (

http://planetmath.org/PseudoOrbit

) in

U

is

δ

-

shadowed

by an orbit of

f

.

Moreover, there is

δ

0

>

0

such that, if

δ

<

δ

0

and if the pseudo-orbit is bi-infinite,
then the shadowing orbit is unique; and if

Λ

has a local

product structure

then the shadowing orbit is in

Λ

.

Title

shadowing lemma

Canonical name

ShadowingLemma

Date of creation

2013-03-22 14:07:22

Last modified on

2013-03-22 14:07:22

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C50

Generated on Fri Feb 9 20:46:41 2018 by

LaTeXML
unstable fixed point

unstable fixed point

A

fixed point

is considered unstable if it is neither attracting nor

Lyapunov stable

. A saddle point is an example of such a fixed point.

Title

unstable fixed point

Canonical

name

UnstableFixedPoint

Date of creation

2013-03-22 13:06:38

Last modified on

2013-03-22 13:06:38

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Definition

Classification

msc 37C70

Related topic

AttractingFixedPoint

Related topic

LyapunovStable

Related topic

StableFixedPoint

Related topic

NeutrallyStableFixedPoint

Related topic

LiapunovStable

Related topic

AsymptoticallyStable

Generated on Fri Feb 9 20:46:42 2018 by

LaTeXML
attracting fixed point

attracting fixed point

Let

X

be a vector field on a manifold

M

and let

F

t

be the flow of

X

. A

fixed point

x

*

of

X

is called

attracting

if there exists a

neighborhood

U

of

x

*

such that for every

x

∈

U

,

F

t

⁢

(

x

)

→

x

*

as

t

→

∞

.

The

stability

of a fixed point can also be classified as stable,

unstable

, neutrally stable, and Liapunov stable.

Title

attracting fixed point

Canonical

name

AttractingFixedPoint

Date of creation

2013-03-22 13:06:24

Last modified on

2013-03-22 13:06:24

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 37C75

Related topic

GloballyAttractingFixedPoint

Related topic

LiapunovStable

Related topic

StableFixedPoint

Related topic

NeutrallyStableFixedPoint

Related topic

UnstableFixedPoint

Generated on Fri Feb 9 20:46:44 2018 by

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Ω-stability theorem

Ω

-stability theorem

Let

M

be a

differentiable manifold

and let

f

:

M

→

M

be a

𝒞

k

diffeomorphism

. We say that

f

is

𝒞

k

-

Ω

-stable, if there is a

neighborhood

𝒰

of

f

in the

𝒞

k

topology

of

Diff

k

⁡

(

M

)

such that for any

g

∈

𝒰

,

f

|

Ω

⁢

(

f

)

is

topologically conjugate

to

g

|

Ω

⁢

(

g

)

.

Ω

-

stability theorem

. If

f

is Axiom A and satisfies the

no-cycles condition

, then

f

is

𝒞

1

-

Ω

-stable.

Remark.

The reciprocal of this theorem is also true (the difficult part is showing that

Ω

-

stability

implies Axiom A), but it is unknown whether

𝒞

k

-

Ω

-stability implies Axiom A when

k

>

1

. This is known as the

𝒞

k

Ω

-stability conjecture

.

Title

Ω

-stability theorem

Canonical name

OmegastabilityTheorem

Date of creation

2013-03-22 14:30:55

Last modified on

2013-03-22 14:30:55

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Theorem

Classification

msc 37C75

Synonym

omega-stability theorem

Synonym

Smale’s

Ω

-stability theorem

Defines

Ω

-stable

Defines

omega-stable

Defines

Ω

-stability

Defines

omega-stability

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

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Γ-equivariant

Γ

-equivariant

Let

Γ

be a

compact Lie group

acting linearly on

V

and let

g

be a mapping defined as

g

:

V

→

V

. Then

g

is

Γ

-

equivariant

if

g

⁢

(

γ

⁢

v

)

=

γ

⁢

g

⁢

(

v

)

for all

γ

∈

Γ

, and all

v

∈

V

.

Therefore if

g

commutes with

Γ

then

g

is

Γ

-equivariant.

[

GSS

]

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in

Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

Γ

-equivariant

Canonical

name

Gammaequivariant

Date of creation

2013-03-22 13:53:20

Last modified on

2013-03-22 13:53:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 37C80

Classification

msc 22-00

Generated on Fri Feb 9 20:46:47 2018 by

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hyperbolic isomorphism

hyperbolic isomorphism

Let

X

be a

Banach space

and

T

:

X

→

X

a

continuous

linear isomorphism

. We say that

T

is an

hyperbolic isomorphism

if its

spectrum

is

disjoint

with the unit circle, i.e.

σ

⁢

(

T

)

∩

{

z

∈

ℂ

:

|

z

|

=

1

}

=

∅

.

If this is the case, by the

spectral theorem

there is a
splitting of

X

into two

invariant subspaces

,

X

=

E

s

⊕

E

u

(and therefore, a corresponding splitting of

T

into two

operators

T

s

:

E

s

→

E

s

and

T

u

:

E

u

→

E

u

, i.e.

T

=

T

s

⊕

T

u

), such that

σ

⁢

(

T

s

)

=

σ

⁢

(

T

)

∩

{

z

:

|

z

|

<

1

}

and

σ

⁢

(

T

u

)

=

σ

⁢

(

T

)

∩

{

z

:

|

z

|

>

1

}

. Also, for any

λ

greater

than the

spectral radius

of both

T

s

and

T

u

-

1

there exists an

equivalent

(box-type) norm

∥

⋅

∥

1

such that

∥

T

s

∥

1

⁢

<

λ

⁢

and

∥

⁢

T

u

-

1

∥

1

<

λ

and

∥

x

∥

1

=

max

⁡

{

∥

x

u

∥

1

,

∥

x

s

∥

1

}

.

In particular,

λ

can be chosen smaller than

1

, so that

T

s

and

T

u

-

1

are

contractions

.

Title

hyperbolic isomorphism

Canonical

name

HyperbolicIsomorphism

Date of creation

2013-03-22 13:39:34

Last modified on

2013-03-22 13:39:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 37D05

Classification

msc 46B03

Synonym

linear hyperbolic isomorphism

Generated on Fri Feb 9 20:46:49 2018 by

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stable manifold

stable manifold

Let

X

be a

topological space

, and

f

:

X

→

X

a

homeomorphism

. If

p

is a

fixed point

for

f

, the

stable and

unstable sets

of

p

are defined by

W

s

⁢

(

f

,

p

)

=

{

q

∈

X

:

f

n

⁢

(

q

)

→

n

→

∞

p

}

,

W

u

⁢

(

f

,

p

)

=

{

q

∈

X

:

f

-

n

⁢

(

q

)

→

n

→

∞

p

}

,

respectively.

If

p

is a periodic point of least period

k

, then it is a fixed point of

f

k

, and the stable and unstable sets of

p

are

W

s

⁢

(

f

,

p

)

=

W

s

⁢

(

f

k

,

p

)

W

u

⁢

(

f

,

p

)

=

W

u

⁢

(

f

k

,

p

)

.

Given a

neighborhood

U

of

p

, the

local stable and unstable sets

of

p

are defined by

W

loc

s

⁢

(

f

,

p

,

U

)

=

{

q

∈

U

:

f

n

⁢

(

q

)

∈

U

⁢

for each

⁢

n

≥

0

}

,

W

loc

u

⁢

(

f

,

p

,

U

)

=

W

loc

s

⁢

(

f

-

1

,

p

,

U

)

.

If

X

is

metrizable

, we can define the stable and unstable sets for any point by

W

s

⁢

(

f

,

p

)

=

{

q

∈

U

:

d

⁢

(

f

n

⁢

(

q

)

,

f

n

⁢

(

p

)

)

→

n

→

∞

0

}

,

W

u

⁢

(

f

,

p

)

=

W

s

⁢

(

f

-

1

,

p

)

,

where

d

is a metric for

X

. This definition clearly
coincides with the previous one when

p

is a periodic point.

When

K

is an invariant subset of

X

, one usually denotes by

W

s

⁢

(

f

,

K

)

and

W

u

⁢

(

f

,

K

)

(or just

W

s

⁢

(

K

)

and

W

u

⁢

(

K

)

) the stable and unstable sets of

K

, defined as the set of points

x

∈

X

such that

d

⁢

(

f

n

⁢

(

x

)

,

K

)

→

0

when

x

→

∞

or

-

∞

, respectively.

Suppose now that

X

is a

compact

smooth manifold

, and

f

is a

𝒞

k

diffeomorphism

,

k

≥

1

. If

p

is a

hyperbolic periodic point

, the

stable manifold theorem

assures that for some neighborhood

U

of

p

, the local stable and unstable sets are

𝒞

k

embedded disks, whose

tangent spaces

at

p

are

E

s

and

E

u

(the stable and

unstable

spaces of

D

⁢

f

⁢

(

p

)

), respectively; moreover, they vary continuously (in certain sense) in a neighborhood of

f

in the

𝒞

k

topology

of

Diff

k

⁡

(

X

)

(the space of all

𝒞

k

diffeomorphisms from

X

to itself). Finally, the stable and unstable sets are

𝒞

k

injectively immersed disks. This is why they are commonly called

stable and unstable

manifolds

. This result is also valid for nonperiodic points, as long as they lie in some

hyperbolic set

(stable manifold theorem for hyperbolic sets).

Title

stable manifold

Canonical name

StableManifold

Date of creation

2013-03-22 13:41:07

Last modified on

2013-03-22 13:41:07

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 37D10

Classification

msc 37C75

Synonym

stable set

Synonym

unstable set

Synonym

unstable manifold

Related topic

HyperbolicFixedPoint

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Anosov diffeomorphism

Anosov diffeomorphism

If

M

is a compact

smooth manifold

, a

diffeomorphism

f

:

M

→

M

(or a flow

ϕ

:

ℝ

×

M

→

M

) such that the whole space

M

is an

hyperbolic set

for

f

(or

ϕ

) is called an

Anosov diffeomorphism

(or flow).

Anosov diffeomorphisms were introduced by D.V. Anosov, who proved that they are

𝒞

1

-

structurally stable

.

Not every

manifold

admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere

S

n

. The simplest examples of

compact manifolds

admiting them are the tori

𝕋

n

: they admit the so called linear Anosov diffeomorphisms, which are

isomorphisms

of

𝕋

n

having no eigenvalue of modulus

1

. It was proved that any other Anosov diffeomorphism in

𝕋

n

is

topologically conjugate

to one of this kind.

It is not known which manifolds

support

Anosov diffeomorphisms. The only known examples of are nilmanifolds and infranilmanifolds, and it is conjectured that these are the only ones.

Anosov flows

are more abundant; for example, if

M

is a

Riemannian manifold

of

negative

sectional curvature

, then its

geodesic flow

is an Anosov flow.

Another famous

conjecture

is that the

nonwandering set

of any Anosov diffeomorphism is the whole manifold

M

. This is known to be true for linear Anosov diffeomorphisms (and hence for any Anosov diffeomorphism in a torus). For Anosov flows, there are examples where the nonwandering set is a

proper subset

of

M

.

Title

Anosov diffeomorphism

Canonical

name

AnosovDiffeomorphism

Date of creation

2013-03-22 13:47:43

Last modified on

2013-03-22 13:47:43

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 37D20

Defines

Anosov flow

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Axiom A

Axiom A

Let

M

be a

smooth manifold

. We say that a

diffeomorphism

f

:

M

→

M

satisfies
(Smale’s)

Axiom A

(or that

f

is an Axiom A diffeomorphism) if

1.

the

nonwandering set

Ω

⁢

(

f

)

has a

hyperbolic structure

;

2.

the set of periodic points of

f

is dense in

Ω

⁢

(

f

)

:

Per

⁡

(

f

)

¯

=

Ω

⁢

(

f

)

.

Sometimes, Axiom A diffeomorphisms are called

hyperbolic diffeomorphisms

, because the portion of

M

where the “interesting” dynamics occur (namely,

Ω

⁢

(

f

)

) has a hyperbolic behaviour.

Title

Axiom A

Canonical name

AxiomA

Date of creation

2013-03-22 13:40:27

Last modified on

2013-03-22 13:40:27

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 37D20

Synonym

hyperbolic diffeomorphism

Generated on Fri Feb 9 20:46:55 2018 by

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hyperbolic set

hyperbolic set

Let

M

be a compact

smooth manifold

, and let

f

:

M

→

M

be a

diffeomorphism

.
An

f

-invariant subset

Λ

of

M

is said to be

hyperbolic

(or to have an hyperbolic structure) if there is a

splitting

of the

tangent bundle

of

M

restricted to

Λ

into a (Whitney) sum of two

D

⁢

f

-invariant subbundles,

E

s

and

E

u

such that the

restriction

of

D

⁢

f

|

E

s

is a

contraction

and

D

⁢

f

|

E

u

is an

expansion

. This means that there are constants

0

<

λ

<

1

and

c

>

0

such that

1.

T

Λ

⁢

M

=

E

s

⊕

E

u

;

2.

D

⁢

f

⁢

(

x

)

⁢

E

x

s

=

E

f

⁢

(

x

)

s

and

D

⁢

f

⁢

(

x

)

⁢

E

x

u

=

E

f

⁢

(

x

)

u

for each

x

∈

Λ

;

3.

∥

D

⁢

f

n

⁢

v

∥

<

c

⁢

λ

n

⁢

∥

v

∥

for each

v

∈

E

s

and

n

>

0

;

4.

∥

D

⁢

f

-

n

⁢

v

∥

<

c

⁢

λ

n

⁢

∥

v

∥

for each

v

∈

E

u

and

n

>

0

.

using some

Riemannian metric

on

M

.

If

Λ

is hyperbolic, then there exists an

adapted

Riemannian metric, i.e. one such that

c

=

1

.

Title

hyperbolic set

Canonical name

HyperbolicSet

Date of creation

2013-03-22 13:40:21

Last modified on

2013-03-22 13:40:21

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37D20

Synonym

hyperbolic structure

Synonym

uniformly hyperbolic

Related topic

HyperbolicFixedPoint

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Smale’s spectral decomposition theorem

Smale’s spectral decomposition theorem

Let

M

be a

compact

differentiable manifold

and let

f

:

M

→

M

be an Axiom A

diffeomorphism

. The

nonwandering set

Ω

of

f

can be partitioned into a finite number of compact

topologically transitive

blocks, called

basic blocks

:

Ω

=

⋃

i

=

1

m

Λ

i

.

Moreover, each basic block is partitioned into a finite number of compact subblocks

Λ

i

⁢

j

,

j

=

1

,

…

,

m

i

such that

f

⁢

(

Λ

i

⁢

j

)

=

Λ

i

⁢

(

j

+

1

)

for

1

≤

j

<

m

i

and

f

⁢

(

Λ

i

⁢

m

i

)

=

Λ

i

⁢

1

, and

Λ

i

⁢

j

is topologically mixing for

f

m

i

.

Title

Smale’s spectral decomposition theorem

Canonical name

SmalesSpectralDecompositionTheorem

Date of creation

2013-03-22 14:28:08

Last modified on

2013-03-22 14:28:08

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Theorem

Classification

msc 37D20

Synonym

spectral decomposition theorem

Defines

basic block

Generated on Fri Feb 9 20:46:59 2018 by

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Kupka-Smale

Kupka-Smale

A

diffeomorphism

f

mapping a

smooth manifold

M

to itself is called a

Kupka-Smale

diffeomorphism if

1.

every periodic point of

f

is hyperbolic;

2.

for each pair of periodic points

p

,

q

of

f

, the intersection between the

stable manifold

of

p

and the unstable manifold of

q

is

transversal

.

Title

Kupka-Smale

Canonical name

KupkaSmale

Date of creation

2013-03-22 13:41:14

Last modified on

2013-03-22 13:41:14

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37D99

Related topic

KupkaSmaleTheorem

Generated on Fri Feb 9 20:47:00 2018 by

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Sharkovskii’s theorem

Sharkovskii’s theorem

Every

natural number

can be written as

2

r

⁢

p

, where

p

is odd, and

r

is the

maximum

exponent

such that

2

r

divides (

http://planetmath.org/

Divisibility

) the given number. We define the

Sharkovskii ordering

of the natural numbers in this way: given two

odd numbers

p

and

q

, and two

nonnegative integers

r

and

s

,
then

2

r

⁢

p

≻

2

s

⁢

q

if

1.

r

<

s

and

p

>

1

;

2.

r

=

s

and

p

<

q

;

3.

r

>

s

and

p

=

q

=

1

.

This defines a

linear ordering

of

ℕ

, in which we first have

3

,

5

,

7

,

…

, followed by

2

⋅

3

,

2

⋅

5

,

…

,
followed by

2

2

⋅

3

,

2

2

⋅

5

,

…

,
and so on, and finally

2

n

+

1

,

2

n

,

…

,

2

,

1

. So it looks like this:

3

≻

5

≻

⋯

≻

3

⋅

2

≻

5

⋅

2

≻

⋯

≻

3

⋅

2

n

≻

5

⋅

2

n

≻

⋯

≻

2

2

≻

2

≻

1

.

Sharkovskii’s

theorem

.

Let

I

⊂

ℝ

be an

interval

, and let

f

:

I

→

ℝ

be a

continuous function

. If

f

has a

periodic point

of least period

n

, then

f

has a periodic point of least period

k

, for each

k

such that

n

≻

k

.

Title

Sharkovskii’s theorem

Canonical name

SharkovskiisTheorem

Date of creation

2013-03-22 13:16:11

Last modified on

2013-03-22 13:16:11

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 37E05

Synonym

Sharkovsky’s theorem

Defines

Sharkovskii’s

ordering

Defines

Sharkovsky’s theorem

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