In mathematics, the orbit capacity of a subset of a

topological dynamical system In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology. Scope The central object of study in topolo ...

may be thought of heuristically as a “topological dynamical

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...

” of the subset. More precisely, its value for a set is a tight upper bound for the normalized number of visits of orbits in this set.

Definition

A topological dynamical system consists of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Hausdorff

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' and a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

T:X\rightarrow X

. Let

E\subset X

be a set. Lindenstrauss introduced the definition of orbit capacity: :

\operatorname(E)=\lim_\sup_ \frac 1 n \sum_^ \chi_E (T^k x)

Here,

\chi_E(x)

is the

membership function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...

for the set

E

. That is

\chi_E(x)=1

x\in E

and is zero otherwise.

Properties

One has

0\le\operatorname(E)\le 1

. By convention, topological dynamical systems do not come equipped with a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Meas ...

; the orbit capacity can be thought of as defining one, in a "natural" way. It is not a true measure, it is only a sub-additive: * Orbit capacity is

sub-additive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two element (set), elements of the Domain of a function, domain always returns something less than or equal to the sum of the ...

: ::

\operatorname(A\cup B)\leq \operatorname(A)+\operatorname(B)

* For a closed set ''C'', ::

\operatorname(C)=\sup_\mu(C)

: Where M_''T''(''X'') is the collection of ''T''- invariant probability measures on ''X''.

Small sets

When

\operatorname(A)=0

A

is called small. These sets occur in the definition of the small boundary property.

References

{{reflist Topological dynamics