Topological Entropy and Mixing Invariant Extremal Distributional Chaos
We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.
On various definitions of shadowing with average error in tracing
When computing a trajectory of a dynamical system, influence of noise can lead to large perturbations which can appear, however, with small probability. Then when calculating approximate trajectories, it makes sense to consider errors small on average, since controlling them in each iteration may be impossible. Demand to relate approximate trajectories with genuine orbits leads to various notions of shadowing (on average) which we consider in the paper. As the main tools in our studies we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center which in turn implies the average shadowing property. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.
The genus Pseudolithophyllum (Corallinaceae) in the Gulf of Maine
SummaryThis is the third of a series of papers covering the anatomy, morphology, systematics and ecology of the crustose corallines of the Gulf of Maine. Pseudolithophyllum orbiculatum, the only species of this genus in the region, is treated in this paper.