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
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Famous quotes containing the word measure:
“...the measure you give will be the measure you get...”
—Bible: New Testament, Mark 4:24.
Jesus.
“Speech is the twin of my vision, it is unequal to measure itself,
It provokes me forever, it says sarcastically,
Walt you contain enough, why don’t you let it out then?”
—Walt Whitman (1819–1892)
“This entire most beautiful order of good things is going to pass away after its measure has been exhausted; for both morning and evening were made in them.”
—St. Augustine (354–430)