A local variational relation and applications

Independence in topological and C*-dynamics

“…By taking δ small enough we can make the unit ball of the operator system W = span{1, h} as close as we wish to the unit ball of V , and so we will obtain (2) once we show that W has a 1-independence set of positive density, and this can be done as in the proof of the corresponding implication in Proposition 4.2 using the partition of unity {h, 1 − h}. In [43] Huang and Ye proposed u.p.e. of all orders as the most suitable topological analogue of a K-system.…”

“…To conclude Section 3 we discuss how the theory readily extends to actions of discrete amenable groups. Note in contrast that the measure-dynamical approach as it has been developed for Z-systems does not directly extend to the general amenable case, as one needs for example to find a substitute for the procedure of taking powers of a single automorphism (see [43] and Section 19.3 of [31]). …”

Entropy points and applications

“…Based on the theories and results established in [2], [5], [15] and [19], in this section we study the structure of the set of entropy points. It is proved that the support of any ergodic measure with positive entropy is contained in the set of entropy points, and the topological entropy of a TDS equals the topological entropy over the set of entropy points whereas it doesn't hold for the set of full entropy points, as E f p (X, T ) could be a singleton for some positive entropy TDS.…”

“…First Blanchard [2] introduced the notion of an entropy pair in topological dynamics from which the existence of the maximal zero entropy factor of a given topological system was proved ( [3]), and Blanchard et al [5] defined the notion of an entropy pair for an invariant measure which naturally led to the discussion of the relation between the two kinds of entropy pairs. Then Huang and Ye [15] introduced the notion of entropy tuples both in topological and measure-theoretical settings. Just recently, Dou, Ye and Zhang [9] generalized the notion (of entropy tuples) to entropy sequences and entropy sets in both settings.…”

“…Just recently, Dou, Ye and Zhang [9] generalized the notion (of entropy tuples) to entropy sequences and entropy sets in both settings. To study the relation of entropy tuples in both settings, a local variational inequality [4], a local variational relation [15] and finally local variational principles [20], [14] which refine the classical variational principle were found. Moreover, a local variational principle for conditional entropy was proved [16], and relative entropy tuples in both settings, relative u.p.e.…”