Li-Yorke Chaos for Composition Operators on LpL^p-Spaces

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on $L^p$ spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp

Expansivity and Shadowing in Linear Dynamics

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators

Graph Theoretic Structure of Maps of the Cantor Space

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of the space of homeomorphisms and the space of self-maps of the Cantor space. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space of homeomorphisms of the Cantor space. The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a comeager subset of homeomorphism space of the Cantor space each element of which has topological entropy zero. Our analogous investigation in the space of continuous self-maps of the Cantor space yields a surprising result: there is a comeager subset of the space of self-maps of the Cantor space such that any two elements of this set are conjugate to each other by an homeomorphism. Our description of this class also yields many old and new results concerning dynamics of a comeager subset of the space of continuous self-maps of the Cantor space.Comment: 26 page

Uniformly Positive Entropy of Induced Transformations

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system (\cM(X),\wt{T}) on the space of Borel probability measures endowed with the weak$^*$ topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.Comment: To apper in ETD

Evolution of Coordination in Social Networks: A Numerical Study

Coordination games are important to explain efficient and desirable social behavior. Here we study these games by extensive numerical simulation on networked social structures using an evolutionary approach. We show that local network effects may promote selection of efficient equilibria in both pure and general coordination games and may explain social polarization. These results are put into perspective with respect to known theoretical results. The main insight we obtain is that clustering, and especially community structure in social networks has a positive role in promoting socially efficient outcomes.Comment: preprint submitted to IJMP

Invasive monitoring of the clinical effects of high intra-abdominal pressure for insertion of the first trocar.

Background: To analyze the effects of transitory, high intra-abdominal pressure on clinical, hemodynamic, blood gas and metabolic parameters.&#xd;&#xa;&#xd;&#xa;Methods: Sixty-seven laparoscopic patients were divided into groups P12 (n = 30, maximum intra-abdominal pressure of 12 mmHg) and P20 (n = 37, maximum intra-abdominal pressure of 20 mmHg). Through radial artery cannulation, mean arterial pressure (MAP) was assessed and blood gas analysis &#x2013; pH, arterial oxygen tension (PaO2), arterial carbon dioxide tension (PaCO2), bicarbonate (HCO3) and base excess (BE) &#x2013; was performed. These parameters were evaluated in both groups at time point zero, before CO2 insufflation; at time point one (TP1), when intra-abdominal pressure of 12 mmHg was reached in both groups; at time point two (TP2), 5 minutes after reaching intra-abdominal pressure of 12 mmHg in group P12 and of 20 mmHg in group P20; and at time point three (TP3), 10 minutes after reaching intra-abdominal pressure of 12 mmHg in group P12 and 10 minutes after TP1 in group P20, when intra-abdominal pressure decreased from 20 mmHg to 12 mmHg. Values out of the normal range or the occurrence of atypical phenomena suggestive of organic disease indicated clinical changes.&#xd;&#xa;&#xd;&#xa;Results: Significant variations in MAP, pH, HCO3 and BE were observed in group P20; these changes, however, were within normal limits. Clinical changes were also within normal limits, and no pathological phenomena were observed.&#xd;&#xa;&#xd;&#xa;Conclusions: Brief, intra-abdominal hypertension for the insertion first trocar insertion causes variations in MAP, pH, HCO3 and BE without adverse effects, and it may protect from iatrogenic injury