Linearized gravity

In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is very much the opposite of a conservative system, for which the ideas of the Poincaré recurrence theorem apply. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park..

Wandering points

A common, discrete-time definition of wandering sets starts with a map $f:X\to X$ of a topological space X. A point $x\in X$ is said to be a wandering point if there is a neighbourhood U of x and a positive integer N such that for all $n>N$, the iterated map is non-intersecting:

$f^n(U)\cap U=\varnothing .$

A handier definition requires only that the intersection have measure zero. To be precise, the definition requires that X be a measure space, i.e. part of a triple $(X,\mathrm{\Sigma },\mu )$ of Borel sets $\mathrm{\Sigma }$ and a measure $\mu$ such that

$\mu (f^n(U)\cap U)=0.$

Similarly, a continuous-time system will have a map ${\phi }_t:X\to X$ defining the time evolution or flow of the system, with the time-evolution operator $\phi$ being a one-parameter continuous abelian group action on X:

${\phi }_{t+s}={\phi }_t\circ {\phi }_s.$

In such a case, a wandering point $x\in X$ will have a neighbourhood U of x and a time T such that for all times $t>T$, the time-evolved map is of measure zero:

$\mu ({\phi }_t(U)\cap U)=0.$

These simpler definitions may be fully generalized to a general group action. Let $\mathrm{\Omega }=(X,\mathrm{\Sigma },\mu )$ be a measure space, that is, a set with a measure defined on its Borel subsets. Let $\mathrm{\Gamma }$ be a group acting on that set. Given a point $x\in \mathrm{\Omega }$, the set

$\{\gamma \cdot x:\gamma \in \mathrm{\Gamma }\}$

is called the trajectory or orbit of the point x.

An element $x\in \mathrm{\Omega }$ is called a wandering point if there exists a neighborhood U of x and a neighborhood V of the identity in $\mathrm{\Gamma }$ such that

$\mu (\gamma \cdot U\cap U)=0$

for all $\gamma \in \mathrm{\Gamma }-V$.

Non-wandering points

The definition for a non-wandering point is in a sense the converse. In the discrete case, $x\in X$ is non-wandering if, for every open set U containing x, one has that

$\mu (f^n(U)\cap U)>0$

for some $n\ge N$ and any $N\ge 1$ arbitrarily large. Similar definitions follow for the continuous-time and discrete and continuous group actions.

Wandering sets and dissipative systems

A wandering set is a collection of wandering points. More precisely, a subset W of $\mathrm{\Omega }$ is a wandering set under the action of a discrete group $\mathrm{\Gamma }$ if W is measurable and if, for any $\gamma \in \mathrm{\Gamma }-\{e\}$ the intersection

$\gamma W\cap W$

is a set of measure zero.

The concept of a wandering set is in a sense dual to the ideas expressed in the Poincaré recurrence theorem. If there exists a wandering set of positive measure, then the action of $\mathrm{\Gamma }$ is said to be dissipative, and the dynamical system $(\mathrm{\Omega },\mathrm{\Gamma })$ is said to be a dissipative system. If there is no such wandering set, the action is said to be conservative, and the system is a conservative system. For example, any system for which the Poincaré recurrence theorem holds cannot have, by definition, a wandering set of positive measure; and is thus an example of a conservative system.

Define the trajectory of a wandering set W as

$W^{*}={\cup }_{\gamma \in \mathrm{\Gamma }}\gamma W.$

The action of $\mathrm{\Gamma }$ is said to be completely dissipative if there exists a wandering set W of positive measure, such that the orbit $W^{*}$ is almost-everywhere equal to $\mathrm{\Omega }$, that is, if

$\mathrm{\Omega }-W^{*}$

is a set of measure zero.

See also

• No wandering domain theorem

References

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