# ℕ_{o} vs. ℤ

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## Abstract

The title of this chapter calls for an explanation: Given a base space *X*, an endomorphism σ of *X*, and a prescribed weight function *W* on *X*, we saw in Section 1.2 that there is an associated measure *P*_{x} on the space of infinite paths rooted at *x*; see Figure 1.1 (p. 8) for an illustration. As noted, the function *W* determines the transition probabilities that go into the probability measure *P*_{x} as follows: For two “successive” points *y* and *z* on such a path in *X*, a transition is possible if σ (*z*)=*y*, and the transition probability is then *W (z)*. If points on the path are further apart, we use a natural formula for conditional probabilities.

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## Copyright information

© Springer Science+Business Media, LLC 2006