Expansive
Encyclopedia
In mathematics
, the notion of expansivity formalizes the notion of points moving away from one-another under the action of an iterated function
. The idea of expansivity is fairly rigid
, as the definition of positive expansivity, below, as well as the Schwarz-Ahlfors-Pick theorem demonstrate.
is a metric space
, a homeomorphism
is said to be expansive if there is a constant
called the expansivity constant, such that for any pair of points
in 
there is an integer 
such that
.
Note that in this definition,
can be positive or negative, and so 
may be expansive in the forward or backward directions.
The space
is often assumed to be compact
, since under that
assumption expansivity is a topological property; i.e. if
is any other metric generating the same topology as 
, and if 
is expansive in 
, then 
is expansive in 
(possibly with a different expansivity constant).
If
is a continuous map, we say that
is positively expansive (or forward expansive) if there is a
such that, for any
in 
, there is an 
such that 
.
and 
there is an 
such that for each pair 
of points of 
such that 
, there is an 
with 
such that
where
is the expansivity constant of 
(proof).
is compact and 
is a positively
expansive homeomorphism, then
is finite (proof).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the notion of expansivity formalizes the notion of points moving away from one-another under the action of an iterated function
Iterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...
. The idea of expansivity is fairly rigid
Rigid
In mathematics, a rigid collection C of mathematical objects is one in which every c ∈ C is uniquely determined by less information about c than one would expect....
, as the definition of positive expansivity, below, as well as the Schwarz-Ahlfors-Pick theorem demonstrate.
Definition
If is a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, a homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
is said to be expansive if there is a constantcalled the expansivity constant, such that for any pair of points
in there is an integer such that.Note that in this definition,
can be positive or negative, and so may be expansive in the forward or backward directions.The space
is often assumed to be compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, since under that
assumption expansivity is a topological property; i.e. if
is any other metric generating the same topology as , and if is expansive in , then is expansive in (possibly with a different expansivity constant).If
is a continuous map, we say that
is positively expansive (or forward expansive) if there is asuch that, for any
in , there is an such that .Theorem of uniform expansivity
Given f an expansive homeomorphism, the theorem of uniform expansivity states that for every and there is an such that for each pair of points of such that , there is an with such thatwhere
is the expansivity constant of (proof).Discussion
Positive expansivity is much stronger than expansivity. In fact, one can prove that if is compact and is a positivelyexpansive homeomorphism, then
is finite (proof).External links
- Expansive dynamical systems on scholarpedia