Subshift of Finite Type - Measure

Measure

A subshift of finite type may be endowed with any one of several different measures, thus leading to a measure-preserving dynamical system. A common object of study is the Markov measure, which is an extension of a Markov chain to the topology of the shift.

A Markov chain is a pair (P,π) consisting of the transition matrix, an matrix for which all and

for all i. The stationary probability vector has all and has

.

A Markov chain, as defined above, is said to be compatible with the shift of finite type if whenever . The Markov measure of a cylinder set may then be defined by

The Kolmogorov-Sinai entropy with relation to the Markov measure is

Read more about this topic:  Subshift Of Finite Type

Famous quotes containing the word measure:

    I candidly confess that I have ever looked on Cuba as the most interesting addition which could ever be made to our system of States. The control which, with Florida, this island would give us over the Gulf of Mexico, and the countries and isthmus bordering on it, as well as all those whose waters flow into it, would fill up the measure of our political well-being.
    Thomas Jefferson (1743–1826)

    I love the pride whose measure is its own eminence and not the insignificance of someone else.
    Franz Grillparzer (1791–1872)

    Perpetual modernness is the measure of merit, in every work of art; since the author of it was not misled by anything short- lived or local, but abode by real and abiding traits.
    Ralph Waldo Emerson (1803–1882)