- Almost sure local limit theorems
- A Breiman Type Theorem for Gibbs Measures - Semantic Scholar
- A Breiman Type Theorem for Gibbs Measures
- A Breiman type theorem for Gibbs measures

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### Almost sure local limit theorems

Request Username Can't sign in? We follow the approach of defining the Hentschel-Procaccia dimension spectrum in [4] and introduce the notion of the Hentschel-Procaccia entropy spectrum. Therefore, by proposition 2. It is a direct consequence of proposition 2.

We consider the particular case of the reference measure m to be a Gibbs measure, and we obtain a formula that connects q-topological entropy and the classical topological pressure. Consequently, we obtain some relations between q-topological and lower and upper q-topological entropies. Now we recall the definition of Gibbs measures.

One can show that h1 s, Z , h1 s, Z and h1 s, Z are respectively the topological pressure and lower and upper topological pressure of a on the set Z see [4] for details. Theorem 2. Analogously, we get the following lower bound of the q-topological entropy. Under the conditions of the above theorem, and using the properties of topological pressure on non-compact subsets see [4] , we obtain the relations between q-topological and lower and upper q-topological entropies.

In this section we introduce different types of q-metric entropy and we study their properties and relations between them. We follow the approach in [4] and introduce the notion of q-metric entropy using the inverse variational principle. We call the quantity hq s, n the q-metric entropy of a with respect to v. We call the quantities hq s, n and hq s, n respectively the lower and upper q-metric entropy of a with respect to v.

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We mention that the q-topological entropy is defined with respect to a fixed reference measure which can be different from v. We shall now show that the q-metric entropy as well as the lower and upper q-metric entropies are invariant under a homeomorphism that respect the measure. Proposition 3. Here the third equality follows from the fact that n is a homeomorphism. The other two equalities for hq and hq can be proven in a similar fashion. The first statement is a direct consequence of statement 3 of proposition 2.

## A Breiman Type Theorem for Gibbs Measures - Semantic Scholar

By remark 2. Since v is ergodic, we have. The last statement follows directly from proposition 2. This completes the proof of the proposition. These two quantities are called local lower and upper metric entropy at u with respect to v respectively. We will prove the first statement; the second one can be proven in a similar fashion.

Fix such a positive integer N. Theorem 3. By the first inequality of 3. We may further assume that C1 is compact, since otherwise we can approximate it from within by a compact subset. This implies that. Given a positive integer N, set. This completes the proof of the theorem. Following the approach in [5], we introduce the modified HP-entropy spectrum.

## A Breiman Type Theorem for Gibbs Measures

Definition 3. The following result gives the relations between modified HP-spectrum for entropies, local lower and upper metric entropy and lower and upper q-metric entropy.

The first statement follows directly from definitions and theorem 2. The second statement is now a direct consequence of the first result and theorem 3. Kieffer, A generalized Shannon—McMillan theorem for the action of an amenable group on a probability space. MathSciNet Google Scholar.

Lindenstrauss, Pointwise theorems for amenable groups.

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## A Breiman type theorem for Gibbs measures

Ornstein and B. Israel J. Ruelle, Thermodynamic formalism. Addison-Wesley, Reading, MA Shulman, Maximal ergodic theorems on groups [in Russian]. Thesis, Vilnius Tempelman, Specific characteristics and variational principle for homogeneous random fields. Wahrscheinlichkeitstheorie und Verwandungen Gebiete 65 , — Kluwer