60 research outputs found
The law of series
We prove a general ergodic-theoretic result concerning the return time
statistic, which, properly understood, sheds some new light on the common sense
phenomenon known as {\it the law of series}. Let \proc be an ergodic process on
finitely many states, with positive entropy. We show that the distribution
function of the normalized waiting time for the first visit to a small cylinder
set is, for majority of such cylinders and up to epsilon, dominated by the
exponential distribution function . This fact has the following
interpretation: The occurrences of such a "rare event" can deviate from
purely random in only one direction -- so that for any length of an
"observation period" of time, the first occurrence of "attracts" its
further repetitions in this period
Isomorphic extensions and applications
If is a topological factor map between uniquely ergodic
topological dynamical systems, then is called an isomorphic extension
of if is also a measure-theoretic isomorphism. We consider the
case when the systems are minimal and we pay special attention to
equicontinuous . We first establish a characterization of this type of
isomorphic extensions in terms of mean equicontinuity, and then show that an
isomorphic extension need not be almost one-to-one, answering questions of Li,
Tu and Ye.Comment: 16 page
Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy
We review subbadditivity properties of Shannon entropy, in particular, from
the Shearer's inequality we derive the "infimum rule" for actions of amenable
groups. We briefly discuss applicability of the "infimum formula" to actions of
other groups. Then we pass to topological entropy of a cover. We prove
Shearer's inequality for disjoint covers and give counterexamples otherwise. We
also prove that, for actions of amenable groups, the supremum over all open
covers of the "infimum fomula" gives correct value of topological entropy.Comment: 12 page
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