# A local variational relation and applications

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## Abstract

In [BGH] the authors show that for a given topological dynamical system (*X,T*) and an open cover*u* there is an invariant measure μ such that inf*h*_{μ}(*T*,ℙ)≥*h*_{top}(*T,U*) where infimum is taken over all partitions finer than*u*. We prove in this paper that if μ is an invariant measure and*h*_{μ}(*T*,ℙ) > 0 for each ℙ finer than*u*, then inf*h*_{μ}(*T*,ℙ > 0 and*h*_{top}(*T,U*) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of order*n* (*n*≥2) or u.p.e. of all orders. We show that for each*n*≥2 the set of all topological entropy*n*-tuples is the union of the set of entropy*n*-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of order*n* and u.p.e. of all orders are obtained.

We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of order*n* does not imply u.p.e. of order*n*+1 for each*n*≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of order*n* (resp. of all orders) systems is again u.p.e. of order*n* (resp. of all orders).

## Keywords

Invariant Measure Topological Entropy Full Support Ergodic Measure Positive Entropy## Preview

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