aarjaneiro/planetmath huggingface dataset:
Maxwell’s equations

Maxwell’s equations

Maxwell’s equations are a set of four

partial differential equations

first combined by James Clerk Maxwell. They may also be written as

integral equations

. Two other important equations, the electromagnetic wave equation and the equation of conservation of charge, may be derived from them.

0.1

Notation

As this article considers merely the mathematical aspects of the equations, natural units have been used throughout. For their use in physics see any classical electromagnetism textbook.

𝐄

=

Electric field strength

𝐁

=

Magnetic flux density

0.2

Gauss’ Law of Electrostatics

∇

⋅

𝐄

=

0

∮

S

𝐄

⋅

d

𝐒

=

0

0.3

Gauss’ Law of Magnetostatics

∇

⋅

𝐁

=

0

∮

S

𝐁

⋅

d

𝐒

=

0

0.4

Faraday’s Law

Differential form

∇

×

𝐄

=

-

∂

⁡

𝐁

∂

⁡

t

Integral form

∮

C

𝐄

⋅

d

𝐥

=

-

d

d

⁢

t

⁢

(

∫

S

𝐁

⋅

d

𝐒

)

0.5

Ampère’s Law

Differential form

∇

×

𝐁

=

∂

⁡

𝐄

∂

⁡

t

Integral form

∮

C

𝐁

⋅

d

𝐥

=

∫

S

∂

⁡

𝐄

∂

⁡

t

⋅

d

𝐒

0.6

Properties of Maxwell’s Equations

These four equations together have several interesting

properties

:

•

Lorentz invariance

•

The fields

𝐄

and

𝐁

may be Helmholtz decomposed into irrotational and solenoidal potentials. A gauge transformation in these

variables

does not affect the values of the fields.

Title

Maxwell’s equations

Canonical

name

MaxwellsEquations

Date of creation

2013-03-22 17:51:34

Last modified on

2013-03-22 17:51:34

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

28

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 35Q60

Classification

msc 78A25

Related topic

PartialDifferentialEquation

Defines

Faraday’s Law

Defines

Ampere’s Law

Defines

Gauss’ Law of Electrostatics

Defines

Gauss’ Law of Magnetostatics

Generated on Fri Feb 9 20:44:37 2018 by

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heat equation

heat equation

The

heat equation

in 1-dimension (for example, along a metal wire) is a

partial differential equation

of the following form:

∂

⁡

u

∂

⁡

t

=

c

2

⋅

∂

2

⁡

u

∂

⁡

x

2

also written as

u

t

=

c

2

⋅

u

x

⁢

x

Where

u

:

ℝ

2

→

ℝ

is the function giving the temperature at time

t

and position

x

and

c

is a real valued constant. This can be easily extended to 2 or 3

dimensions

as

u

t

=

c

2

⋅

(

u

x

⁢

x

+

u

y

⁢

y

)

and

u

t

=

c

2

⋅

(

u

x

⁢

x

+

u

y

⁢

y

+

u

z

⁢

z

)

Note that in the steady state, that is when

u

t

=

0

, we are left with the

Laplacian

of

u

:

Δ

⁢

u

=

0

Title

heat equation

Canonical

name

HeatEquation

Date of creation

2013-03-22 12:45:36

Last modified on

2013-03-22 12:45:36

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 35Q99

Related topic

DifferentialEquation

Related topic

Laplacian

Generated on Fri Feb 9 20:44:42 2018 by

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no-cycles condition

no-cycles condition

Let

X

be a

metric space

and let

f

:

X

→

X

be a

homeomorphism

.
Suppose

ℱ

=

{

Λ

1

,

…

,

Λ

k

}

is a family of

compact

invariant sets for

f

. Define a

relation

→

on

ℱ

by

Λ

i

→

Λ

j

if

W

u

⁢

(

Λ

i

)

∩

W

s

⁢

(

Λ

j

)

-

⋃

l

=

1

k

Λ

l

≠

∅

,

that is, if the

unstable set

of

Λ

i

intersects the

stable set

of

Λ

j

outside the union of the

Λ

l

’s.

A cycle for

ℱ

is a sequence

{

n

i

:

i

=

1

,

…

,

j

}

such that

Λ

n

i

→

Λ

n

i

+

1

for

1

≤

i

<

j

and

Λ

n

j

→

Λ

n

1

.

With some abuse of notation, we can write this as

Λ

n

1

→

Λ

n

2

→

⋯

→

Λ

n

j

→

Λ

n

1

.

If

ℱ

has no cycles, then we say that it satisfies the

no-cycles condition

.

Title

no-cycles condition

Canonical name

NocyclesCondition

Date of creation

2013-03-22 14:30:53

Last modified on

2013-03-22 14:30:53

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 37-00

Classification

msc 37C75

Synonym

no-cycles

Synonym

no-cycle

Synonym

no cycles condition

Synonym

cycle

Generated on Fri Feb 9 20:45:02 2018 by

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symbolic dynamics

symbolic dynamics

In mathematics,

symbolic dynamics

is the practice of modelling a

dynamical system

by a space consisting of

infinite

sequences of abstract symbols, each symbol corresponding to a state of the system, and a shift operator corresponding to the dynamics. Symbolic dynamics was first introduced by Emil Artin in 1924, in the study of Artin billiards.

Symbolic dynamics originated as a method to study general dynamical systems, but its techniques and ideas have found significant applications in data storage and transmission, linear algebra, the motions of the planets and many other areas. The distinct feature in symbolic dynamics is that time is measured in discrete intervals. So at each time interval the system is in a particular state. Each state is associated with a symbol and the evolution of the system is described by an infinite sequence of symbols - represented effectively as strings. If the system states are not inherently discrete, then the state vector must be discretized, so as to get a coarse-grained description of the system.

This entry was adapted from the Wikipedia article

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

Symbolic dynamics as of December 19, 2006.

Title

symbolic dynamics

Canonical name

SymbolicDynamics

Date of creation

2013-03-22 16:28:29

Last modified on

2013-03-22 16:28:29

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 37-00

Generated on Fri Feb 9 20:45:09 2018 by

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continuous epimorphism of compact groups preserves Haar measure

continuous epimorphism of compact groups preserves Haar measure

\PMlinkescapephrase

right

\PMlinkescapephrase

preserves

Theorem

-

Let

G

,

H

be

compact

Hausdorff

topological groups

. If

ϕ

:

G

⟶

H

is a

continuous

surjective

homomorphism

, then

ϕ

is a

measure preserving

transformation

, in the sense that it preserves the normalized

Haar measure

.

:

Let

μ

be the Haar measure in

G

(normalized, i.e.

μ

⁢

(

G

)

=

1

). Let

ν

be defined for measurable subsets

E

of

H

by

ν

⁢

(

E

)

=

μ

⁢

(

ϕ

-

1

⁢

(

E

)

)

It is easy to see that

ν

defines a

measure

in

H

. Let us now see that

ν

is

invariant

under right translations. For every

s

∈

G

and every measurable subset

E

⊂

H

we have that

ϕ

-

1

⁢

(

ϕ

⁢

(

s

)

⁢

E

)

=

s

⁢

ϕ

-

1

⁢

(

E

)

(1)

The

inclusion

⊇

is obvious. To prove the other inclusion notice that if

z

∈

ϕ

-

1

⁢

(

ϕ

⁢

(

s

)

⁢

E

)

then

ϕ

⁢

(

z

)

=

ϕ

⁢

(

s

)

⁢

t

for some

t

∈

E

. Hence,

ϕ

⁢

(

s

-

1

⁢

z

)

=

t

, i.e

s

-

1

⁢

z

∈

ϕ

-

1

⁢

(

E

)

. It now follows that

z

=

s

⁢

(

s

-

1

⁢

z

)

∈

s

⁢

ϕ

-

1

⁢

(

E

)

.

Thus, equality (1) and the fact that

μ

is a Haar measure imply that

ν

⁢

(

ϕ

⁢

(

s

)

⁢

E

)

=

μ

⁢

(

ϕ

-

1

⁢

(

ϕ

⁢

(

s

)

⁢

E

)

)

=

μ

⁢

(

s

⁢

ϕ

-

1

⁢

(

E

)

)

=

μ

⁢

(

ϕ

-

1

⁢

(

E

)

)

=

ν

⁢

(

E

)

Since

ϕ

is surjective it follows that

ν

is right invariant. It is not difficult to see that

ν

is

regular

, finite on compact sets and

ν

⁢

(

H

)

=

1

. Hence,

ν

is the normalized Haar measure in

H

and, by definition, we have that

ν

⁢

(

E

)

=

μ

⁢

(

ϕ

-

1

⁢

(

E

)

)

Thus,

ϕ

preserves the Haar measure.

□

Title

continuous epimorphism of compact groups preserves Haar measure

Canonical name

ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure

Date of creation

2013-03-22 17:59:06

Last modified on

2013-03-22 17:59:06

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 37A05

Classification

msc 28C10

Classification

msc 22C05

Generated on Fri Feb 9 20:45:12 2018 by

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Krylov-Bogolubov theorem

Krylov-Bogolubov theorem

Any

continuous map

on a

metrizable

compact

topological space

has an invariant Borel probability measure (i.e. a Borel probability measure that is preserved by the given map).

Title

Krylov-Bogolubov theorem

Canonical name

KrylovBogolubovTheorem

Date of creation

2013-03-22 14:07:10

Last modified on

2013-03-22 14:07:10

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Theorem

Classification

msc 37A05

Classification

msc 28C15

Synonym

existence of invariant measures

Synonym

Krylov-Bogoliubov

Generated on Fri Feb 9 20:45:14 2018 by

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proof of Poincaré recurrence theorem 1

proof of Poincaré recurrence theorem 1

Let

A

n

=

∪

k

=

n

∞

f

-

k

⁢

E

. Clearly,

E

⊂

A

0

and

A

i

⊂

A

j

when

j

≤

i

. Also,

A

i

=

f

j

-

i

⁢

A

j

, so that

μ

⁢

(

A

i

)

=

μ

⁢

(

A

j

)

for all

i

,

j

≥

0

, by
the

f

-invariance of

μ

.
Now for any

n

>

0

we have

E

-

A

n

⊂

A

0

-

A

n

, so that

μ

⁢

(

E

-

A

n

)

≤

μ

⁢

(

A

0

-

A

n

)

=

μ

⁢

(

A

0

)

-

μ

⁢

(

A

n

)

=

0

.

Hence

μ

⁢

(

E

-

A

n

)

=

0

for all

n

>

0

, so that

μ

(

E

-

∩

n

=

1

∞

A

n

)

=

μ

(

∪

n

=

1

∞

E

-

A

n

)

=

0

.
But

E

-

∩

n

=

1

∞

A

n

is precisely the set of those

x

∈

E

such
that for some

n

and for all

k

>

n

we have

f

k

⁢

(

x

)

∉

E

.

□

Title

proof of Poincaré recurrence theorem 1

Canonical name

ProofOfPoincareRecurrenceTheorem1

Date of creation

2013-03-22 14:29:56

Last modified on

2013-03-22 14:29:56

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Proof

Classification

msc 37A05

Classification

msc 37B20

Generated on Fri Feb 9 20:45:17 2018 by

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proof of Poincaré recurrence theorem 2

proof of Poincaré recurrence theorem 2

Let

{

U

n

:

n

∈

ℕ

}

be a basis of

open sets

for

X

, and for each

n

define

U

n

′

=

{

x

∈

U

n

:

∀

n

≥

1

,

f

n

⁢

(

x

)

∉

U

n

}

.

From theorem 1 we know that

μ

⁢

(

U

n

′

)

=

0

. Let

N

=

⋃

n

∈

ℕ

U

n

′

.

Then

μ

⁢

(

N

)

=

0

. We assert that if

x

∈

X

-

N

then

x

is recurrent. In fact,
given a

neighborhood

U

of

x

, there is a basic neighborhood

U

n

such that

x

⊂

U

n

⊂

U

, and since

x

∉

N

we have that

x

∈

U

n

-

U

n

′

which by definition of

U

n

′

means that there exists

n

≥

1

such that

f

n

⁢

(

x

)

∈

U

n

⊂

U

; thus

x

is recurrent.

□

Title

proof of Poincaré recurrence theorem 2

Canonical name

ProofOfPoincareRecurrenceTheorem2

Date of creation

2013-03-22 14:29:58

Last modified on

2013-03-22 14:29:58

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Proof

Classification

msc 37A05

Classification

msc 37B20

Generated on Fri Feb 9 20:45:19 2018 by

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mixing

mixing

Let

f

be a

measure-preserving transformation

of a

probability space

(

X

,

𝒜

,

μ

)

. We say that

f

is

mixing

(or

strong-mixing

) if for all

A

,

B

∈

𝒜

,

lim

n

→

∞

⁡

μ

⁢

(

f

-

n

⁢

(

A

)

∩

B

)

=

μ

⁢

(

A

)

⁢

μ

⁢

(

B

)

,

and

f

is

weakly mixing

if

lim

n

→

∞

⁡

1

n

⁢

∑

i

=

0

n

-

1

|

μ

⁢

(

f

-

i

⁢

(

A

)

∩

B

)

-

μ

⁢

(

A

)

⁢

μ

⁢

(

B

)

|

=

0

for all

A

,

B

∈

𝒜

.

Every mixing

transformation

is weakly mixing, and every weakly mixing transformation is

ergodic

.

Title

mixing

Canonical name

Mixing

Date of creation

2013-03-22 14:06:34

Last modified on

2013-03-22 14:06:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 37A25

Defines

strongly mixing

Defines

strong mixing

Defines

strong-mixing

Defines

weak-mixing

Defines

weakly mixing

Defines

weak mixing

Generated on Fri Feb 9 20:45:22 2018 by

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ergodic theorem

ergodic theorem

Let

(

X

,

𝔅

,

μ

)

be a

probability space

,

f

∈

L

1

⁢

(

μ

)

, and

T

:

X

→

X

a

measure preserving

transformation

. Birkhoff’s

ergodic theorem

(often called the

pointwise

or

strong

ergodic theorem) states that there exists

f

*

∈

L

1

⁢

(

μ

)

such that

lim

n

→

∞

⁡

1

n

⁢

∑

k

=

0

n

-

1

f

⁢

(

T

k

⁢

x

)

=

f

*

⁢

(

x

)

for almost all

x

∈

X

. Moreover,

f

*

is

T

-invariant (i.e.,

f

*

∘

T

=

f

*

) almost everywhere and

∫

f

*

⁢

𝑑

μ

=

∫

f

⁢

𝑑

μ

.

In particular, if

T

is

ergodic

then the

T

-invariance of

f

*

implies that it is constant almost everywhere, and so this constant must be the

integral

of

f

*

; that is, if

T

is ergodic, then

lim

n

→

∞

⁡

1

n

⁢

∑

k

=

0

n

-

1

f

⁢

(

T

k

⁢

x

)

=

∫

f

⁢

𝑑

μ

for almost every

x

. This is often interpreted in the following way: for an

ergodic transformation

, the time average equals the space average almost surely.

Title

ergodic theorem

Canonical name

ErgodicTheorem

Date of creation

2013-03-22 12:20:52

Last modified on

2013-03-22 12:20:52

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Theorem

Classification

msc 37A30

Classification

msc 47A35

Synonym

strong ergodic theorem

Synonym

Birkhoff ergodic theorem

Synonym

Birkhoff-Khinchin ergodic theorem

Related topic

ErgodicTransformation

Generated on Fri Feb 9 20:45:24 2018 by

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fundamental theorem of demography

fundamental theorem of demography

Let

A

t

be a sequence of

n

×

n

nonnegative

primitive matrices

. Suppose that

A

t

→

A

∞

, with

A

∞

also a nonnegative primitive matrix. Define the sequence

x

t

+

1

=

A

t

⁢

x

t

, with

x

t

∈

ℝ

n

.
If

x

0

≥

0

, then

lim

t

→

∞

⁡

x

t

∥

x

t

∥

=

p

where

p

is the normalized (

∥

p

∥

=

1

)

eigenvector

associated to the

eigenvalue

of

A

∞

(also called the Perron-Frobenius eigenvector of

A

∞

).

Title

fundamental theorem of demography

Canonical name

FundamentalTheoremOfDemography

Date of creation

2013-03-22 13:18:28

Last modified on

2013-03-22 13:18:28

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

6

Author

jarino (552)

Entry type

Theorem

Classification

msc 37A30

Classification

msc 92D25

Synonym

a weak ergodic theorem

Related topic

PerronFrobeniusTheorem

Generated on Fri Feb 9 20:45:25 2018 by

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discontinuous action

discontinuous action

Let

X

be a

topological space

and

G

a group that acts on

X

by

homeomorphisms

. The action of

G

is said to be

discontinuous

at

x

∈

X

if there is a

neighborhood

U

of

x

such
that the set

{

g

∈

G

|

g

⁢

U

∩

U

≠

∅

}

is finite. The action is called discontinuous if it is discontinuous at
every point.

Remark 1

.

If

G

acts discontinuously then the orbits of the action have no

accumulation points

, i.e. if

{

g

n

}

is a sequence of distinct elements of

G

and

x

∈

X

then the sequence

{

g

n

⁢

x

}

has no

limit points

. If

X

is

locally compact

then an action that satisfies this condition is discontinuous.

Remark 2

.

Assume that

X

is a

locally compact Hausdorff space

and let

Aut

⁡

(

X

)

denote the group of self homeomorphisms of

X

endowed with the

compact-open topology

.
If

ρ

:

G

→

Aut

⁡

(

X

)

defines a discontinuous action
then the image

ρ

⁢

(

G

)

is a

discrete subset

of

Aut

⁡

(

X

)

.

Title

discontinuous action

Canonical

name

DiscontinuousAction

Date of creation

2013-03-22 13:28:49

Last modified on

2013-03-22 13:28:49

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

7

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 37B05

Related topic

PoperlyDiscontinuousAction

Defines

discontinuous action

Generated on Fri Feb 9 20:45:27 2018 by

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mixing action

mixing action

Let

X

be a

topological space

and let

G

be a semigroup.
An action

ϕ

=

{

ϕ

g

}

g

∈

G

of

G

on

X

is (topologically) mixing
if, given any two

open

subsets

U

,

V

of

X

,
the

intersection

U

∩

ϕ

g

⁢

(

V

)

is nonempty
for all

g

∈

G

except at most finitely many.

Example 1.

Let

F

:

X

→

X

be a

continuous function

.
Then

F

is

topologically mixing

if and only if
the action of the monoid

ℕ

on

X

defined by

ϕ

n

⁢

(

x

)

=

F

n

⁢

(

x

)

is mixing according to the definition given above.

Example 2.

Suppose

X

is a

discrete

nonempty set

and

G

is a group;
endow

X

G

with the

product topology

.
The action of

G

on

X

G

defined by

ϕ

g

⁢

(

c

)

⁢

(

z

)

=

c

⁢

(

g

⋅

z

)

⁢

∀

c

:

G

→

X

is mixing.

To prove this fact,
we may suppose without loss of generality
that

U

and

V

are two

cylindric sets

of the form:

U

=

{

c

∈

X

G

∣

c

|

E

=

u

|

E

}

V

=

{

c

∈

X

G

∣

c

|

F

=

v

|

F

}

for suitable finite subsets

E

,

F

⊆

G

and

functions

u

,

v

:

G

→

X

.
Then the only chance for

U

∩

ϕ

g

⁢

(

V

)

to be empty,
is that

e

=

g

⁢

f

for some

e

∈

E

,

f

∈

F

such that

u

⁢

(

e

)

≠

v

⁢

(

f

)

:
but then,

g

∈

E

⁢

F

-

1

, which is finite.

Title

mixing action

Canonical name

MixingAction

Date of creation

2013-03-22 19:19:32

Last modified on

2013-03-22 19:19:32

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 37B05

Generated on Fri Feb 9 20:45:29 2018 by

LaTeXML
Garden of Eden

Garden of Eden

A

Garden of Eden

(briefly,

GoE

)
for a

cellular automaton

𝒜

=

⟨

Q

,

𝒩

,

f

⟩

on a group

G

is a

configuration

c

∈

Q

G

which is not in the image of the global

function

F

𝒜

of

𝒜

.

In other words, a Garden of Eden is a global situation
which can be started from, but never returned to.

The finitary counterpart of a

GoE

configuration
is an

orphan pattern

:
a pattern which cannot be obtained
by synchronous application of the local function

f

.

Of course,
any cellular automaton with an orphan pattern
also has a

GoE

configuration.

Lemma 1

(Orphan pattern principle)

If a cellular automaton with

finite set

of states

Q

has a

GoE

configuration,
then it also has an orphan pattern.

Proof.

First, suppose that

G

is

countable

.
Let

𝒜

=

⟨

Q

,

𝒩

,

f

⟩

be a cellular automaton with no orphan pattern.
Let

c

:

G

→

Q

be a configuration:
we will prove that there is some

e

:

G

→

Q

such that

F

𝒜

⁢

(

e

)

=

c

.

Let

G

=

{

g

n

}

n

≥

0

be an enumeration of

G

:
put

E

n

=

{

g

i

∣

i

≤

n

}

and let

p

n

:

E

n

→

Q

be defined as

p

n

=

c

|

E

n

.

By

hypothesis

, none of the

p

n

’s is an orphan,
so there is a

sequence

of configurations

c

n

:

G

→

Q

satisfying

F

𝒜

⁢

(

c

n

)

|

E

n

=

p

n

=

c

|

E

n

.

It is easy to see that

lim

k

→

∞

⁡

F

𝒜

⁢

(

c

n

k

)

=

c

.

But if

Q

is finite,
then

Q

G

is compact by Tychonoff’s

theorem

,
so there exista a

subsequence

{

c

n

k

}

k

≥

0

and a configuration

e

:

G

→

Q

satisfying

lim

k

→

∞

⁡

c

n

k

=

e

:

Since

F

𝒜

is

continuous

in the product topology,

F

⁢

(

e

)

=

c

.

Let now

G

be arbitrary.
Let

H

be the

subgroup

generated by the

neighborhood

index

𝒩

:
since

𝒩

is finite,

H

is countable.
Let

J

be a set of representatives of the

left cosets

of

H

in

G

,
so that

G

=

⊔

j

∈

J

j

⁢

H

.

(Observe that we do

not

require that

H

is normal in

G

.)
Call

𝒜

H

the cellular automaton

on

H

that has the same local description
(set of states, neighborhood index, local function) as

𝒜

.
Let

c

:

G

→

Q

be a Garden of Eden configuration for

𝒜

:
then at least one of the configurations

c

j

⁢

(

h

)

=

c

⁢

(

j

⁢

h

)

must be a Garden of Eden for

𝒜

H

.
By the discussion above,

𝒜

H

must have an orphan pattern,
which is also an orphan pattern for

𝒜

.

□

Title

Garden of Eden

Canonical name

GardenOfEden

Date of creation

2013-03-22 19:22:01

Last modified on

2013-03-22 19:22:01

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

6

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 37B15

Classification

msc 68Q80

Defines

orphan pattern (cellular automaton)

Defines

orphan pattern principle

Generated on Fri Feb 9 20:45:37 2018 by

LaTeXML
nonwandering set

nonwandering set

Let

X

be a

metric space

, and

f

:

X

→

X

a

continuous

surjection

.
An

element

x

of

X

is a

wandering point

if there is a

neighborhood

U

of

x

and an integer

N

such that, for all

n

≥

N

,

f

n

⁢

(

U

)

∩

U

=

∅

. If

x

is not wandering, we call it a

nonwandering point

. Equivalently,

x

is a nonwandering point if for every neighborhood

U

of

x

there is

n

≥

1

such that

f

n

⁢

(

U

)

∩

U

is nonempty. The set of all nonwandering points is called the

nonwandering set

of

f

, and is denoted by

Ω

⁢

(

f

)

.

If

X

is

compact

, then

Ω

⁢

(

f

)

is compact, nonempty, and forward

invariant

; if, additionally,

f

is an

homeomorphism

, then

Ω

⁢

(

f

)

is invariant.

Title

nonwandering set

Canonical name

NonwanderingSet

Date of creation

2013-03-22 13:39:31

Last modified on

2013-03-22 13:39:31

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 37B20

Related topic

OmegaLimitSet3

Related topic

RecurrentPoint

Defines

wandering point

Defines

nonwandering point

Generated on Fri Feb 9 20:45:42 2018 by

LaTeXML
Poincaré recurrence theorem

Poincaré recurrence theorem

Let

(

X

,

𝒮

,

μ

)

be a

probability space

and let

f

:

X

→

X

be a

measure preserving

transformation

.

Theorem 1

.

For any

E

∈

S

,
the set of those points

x

of

E

such that

f

n

⁢

(

x

)

∉

E

for all

n

>

0

has zero

measure

. That is, almost every point of

E

returns to

E

. In fact, almost every point returns

infinitely often

; i.e.

μ

⁢

(

{

x

∈

E

:

there exists

⁢

N

⁢

such that

⁢

f

n

⁢

(

x

)

∉

E

⁢

for all

⁢

n

>

N

}

)

=

0

.

The following is a topological version of this

theorem

:

Theorem 2

.

If

X

is a

second countable

Hausdorff space

and

S

contains the Borel sigma-algebra, then the
set of

recurrent points

of

f

has full measure. That is, almost every
point is recurrent.

Title

Poincaré recurrence theorem

Canonical name

PoincareRecurrenceTheorem

Date of creation

2013-03-22 14:29:50

Last modified on

2013-03-22 14:29:50

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Theorem

Classification

msc 37B20

Classification

msc 37A05

Generated on Fri Feb 9 20:45:44 2018 by

LaTeXML
recurrent point

recurrent point

Let

X

be a

Hausdorff space

and

f

:

X

→

X

a

function

. A point

x

∈

X

is said to be

recurrent

(for

f

) if

x

∈

ω

⁢

(

x

)

, i.e. if

x

belongs to its

ω

-limit (

http://planetmath.org/OmegaLimitSet3

) set. This means that for each

neighborhood

U

of

x

there exists

n

>

0

such that

f

n

⁢

(

x

)

∈

U

.

The

closure

of the set of

recurrent points

of

f

is often denoted

R

⁢

(

f

)

and is called the

recurrent set

of

f

.

Every recurrent point is a

nonwandering point

, hence if

f

is a

homeomorphism

and

X

is

compact

,

R

⁢

(

f

)

is an invariant subset of

Ω

⁢

(

f

)

, which may be a

proper subset

.

Title

recurrent point

Canonical name

RecurrentPoint

Date of creation

2013-03-22 14:29:53

Last modified on

2013-03-22 14:29:53

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 37B20

Related topic

NonwanderingSet

Defines

recurrent set

Generated on Fri Feb 9 20:45:46 2018 by

LaTeXML