Markov partition explained
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like.
Motivation
Let
be a discrete dynamical system. A basic method of studying its dynamics is to find a
symbolic representation: a faithful encoding of the points of
by sequences of symbols such that the map
becomes the
shift map.
Suppose that
has been divided into a number of pieces
which are thought to be as small and localized, with virtually no overlaps. The behavior of a point
under the iterates of
can be tracked by recording, for each
, the part
which contains
. This results in an infinite sequence on the alphabet
which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of
becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a
shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system
.
Formal definition
A Markov partition[1] is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles
such that
, that
\operatorname{Int}Ei\cap\operatorname{Int}Ej=\emptyset
for
and
\varphi(x)\in\operatorname{Int}Ej
, then
\varphi\left[Wu(x)\capEi\right]\supsetWu(\varphix)\capEj
\varphi\left[Ws(x)\capEi\right]\subsetWs(\varphix)\capEj
Here,
and
are the unstable and
stable manifolds of
x, respectively, and
simply denotes the interior of
. These last two conditions can be understood as a statement of the
Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a
Markov shift; that this is indeed the case is due to theorems by
Yakov Sinai (1968)
[2] and
Rufus Bowen (1975),
[3] thus putting symbolic dynamics on a firm footing.
Variants of the definition are found, corresponding to conditions on the geometry of the pieces
.
[4] Examples
Markov partitions have been constructed in several situations.
Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe.
The system
has the Markov partition
, and in this case the symbolic representation of a real number in
is its binary expansion. For example:
x\inE0,Tx\inE1,T2x\inE1,T3x\inE1,T4x\inE0 ⇒ x=(0.01110...)2
. The assignment of points of
to their sequences in the Markov partition is well defined except on the dyadic rationals - morally speaking, this is because
(0.01111...)2=(0.10000...)2
, in the same way as
in decimal expansions.
References
- Book: Douglas . Lind . Brian . Marcus . An introduction to symbolic dynamics and coding . . 1995 . 978-0-521-55124-3 . 1106.37301 .
- Book: Pytheas Fogg, N. . Berthé, Valérie. Valérie Berthé. Ferenczi, Sébastien. Mauduit, Christian. Siegel, Anne . Substitutions in dynamics, arithmetics and combinatorics . Lecture Notes in Mathematics . 1794 . Berlin . . 2002 . 978-3-540-44141-0 . 1014.11015 .
Notes and References
- Book: Gaspard, Pierre . Chaos, scattering and statistical mechanics . Cambridge Nonlinear Science Series . 9 . Cambridge . . 1998 . 978-0-521-39511-3 . 0915.00011 .
- . .
- Pytheas Fogg (2002), p. 208.
- Pytheas Fogg (2002), p. 206.