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Register a new userWhen is a dynamical system mean sensitive?
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This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.
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It is shown that any non-PI minimal system is Li-Yorke sensitive. Consequently, any minimal system with nontrivial weakly mixing factor (such a system is non-PI) is Li-Yorke sensitive, which answers affirmatively an open question by Akin and Kolyada.
We consider the simplest non-linear discrete dynamical systems, given by the logistic maps $f_{a}(x)=ax(1-x)$ of the interval $[0,1]$. We show that there exist real parameters $ain (0,4)$ for which almost every orbit of $f_a$ has the same statistical distribution in $[0,1]$, but this limiting distribution is not Turing computable. In particular, the Monte Carlo method cannot be applied to study these dynamical systems.
We consider the problem of stabilizing an undisturbed, scalar, linear system over a timing channel, namely a channel where information is communicated through the timestamps of the transmitted symbols. Each symbol transmitted from a sensor to a controller in a closed-loop system is received subject to some to random delay. The sensor can encode messages in the waiting times between successive transmissions and the controller must decode them from the inter-reception times of successive symbols. This set-up is analogous to a telephone system where a transmitter signals a phone call to a receiver through a ring and, after the random delay required to establish the connection, the receiver is aware of the ring being received. Since there is no data payload exchange between the sensor and the controller, the set-up provides an abstraction for performing event-triggering control with zero payload rate. We show the following requirement for stabilization: for the state of the system to converge to zero in probability, the timing capacity of the channel should be at least as large as the entropy rate of the system. Conversely, in the case the symbol delays are exponentially distributed, we show a tight sufficient condition using a coding strategy that refines the estimate of the decoded message every time a new symbol is received. Our results generalize previous event-triggering control approaches, revealing a fundamental limit in using timing information for stabilization, independent of any transmission strategy.
We consider metrizable ergodic topological dynamical systems over locally compact, $sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.
Numerous studies and anecdotes demonstrate the wisdom of the crowd, the surprising accuracy of a groups aggregated judgments. Less is known, however, about the generality of crowd wisdom. For example, are crowds wise even if their members have systematic judgmental biases, or can influence each other before members render their judgments? If so, are there situations in which we can expect a crowd to be less accurate than skilled individuals? We provide a precise but general definition of crowd wisdom: A crowd is wise if a linear aggregate, for example a mean, of its members judgments is closer to the target value than a randomly, but not necessarily uniformly, sampled member of the crowd. Building on this definition, we develop a theoretical framework for examining, a priori, when and to what degree a crowd will be wise. We systematically investigate the boundary conditions for crowd wisdom within this framework and determine conditions under which the accuracy advantage for crowds is maximized. Our results demonstrate that crowd wisdom is highly robust: Even if judgments are biased and correlated, one would need to nearly deterministically select only a highly skilled judge before an individuals judgment could be expected to be more accurate than a simple averaging of the crowd. Our results also provide an accuracy rationale behind the need for diversity of judgments among group members. Contrary to folk explanations of crowd wisdom which hold that judgments should ideally be independent so that errors cancel out, we find that crowd wisdom is maximized when judgments systematically differ as much as possible. We re-analyze data from two published studies that confirm our theoretical results.
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