Generic points of shift-invariant measures in the countable symbolic space

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving th...

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 2019-03, Vol.166 (2), p.381-413
Main Authors: FAN, AI–HUA, LI, MING–TIAN, MA, JI–HUA
Format: Article
Language:English
Subjects:
Convergence
Entropy
Invariants
Mathematics
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Summary:We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004118000038