Geometry and Topology of Escape II: Homotopic Lobe Dynamics

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an ``Epistrophe Start Rule'': a new epistrophe is spawned Delta = D+1 iterates after the segment to which it converges, where D is the minimum delay time of the complex.Comment: 13 pages, 8 figures, to appear in Chaos, second of two paper

Geometry and Topology of Escape I: Epistrophes

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an ``escape-time plot''. For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called ``epistrophes'', which occur at all levels of resolution within the escape-time plot. (The word ``epistrophe'' comes from rhetoric and means ``a repeated ending following a variable beginning''.) The epistrophes give the escape-time plot a certain self-similarity, called ``epistrophic'' self-similarity, which need not imply either strict or asymptotic self-similarity.Comment: 15 pages, 9 figures, to appear in Chaos, first of two paper

Constraining recent lead pollution sources in the North Pacific using ice core stable lead isotopes

Trends and sources of lead (Pb) aerosol pollution in the North Pacific rim of North America from 1850 to 2001 are investigated using a high-resolution (subannual to annual) ice core record recovered from Eclipse Icefield (3017 masl; St. Elias Mountains, Canada). Beginning in the early 1940s, increasing Pb concentration at Eclipse Icefield occurs coevally with anthropogenic Pb deposition in central Greenland, suggesting that North American Pb pollution may have been in part or wholly responsible in both regions. Isotopic ratios (208Pb/207Pb and 206Pb/207Pb) from 1970 to 2001 confirm that a portion of the Pb deposited at Eclipse Icefield is anthropogenic, and that it represents a variable mixture of East Asian (Chinese and Japanese) emissions transported eastward across the Pacific Ocean and a North American component resulting from transient meridional atmospheric flow. Based on comparison with source material Pb isotope ratios, Chinese and North American coal combustion have likely been the primary sources of Eclipse Icefield Pb over the 1970–2001 time period. The Eclipse Icefield Pb isotope composition also implies that the North Pacific mid-troposphere is not directly impacted by transpolar atmospheric flow from Europe. Annually averaged Pb concentrations in the Eclipse Icefield ice core record show no long-term trend during 1970–2001; however, increasing 208Pb/207Pb and decreasing 206Pb/207Pb ratios reflect the progressive East Asian industrialization and increase in Asian pollutant outflow. The post-1970 decrease in North American Pb emissions is likely necessary to explain the Eclipse Icefield Pb concentration time series. When compared with low (lichen) and high (Mt. Logan ice core) elevation Pb data, the Eclipse ice core record suggests a gradual increase in pollutant deposition and stronger trans-Pacific Asian contribution with rising elevation in the mountains of the North Pacific rim

The excess choice effect:The role of outcome valence and counterfactual thinking

Contrary to economic theory, psychological research has demonstrated increased choice can undermine satisfaction. When and why this 'excess choice effect' (ECE) occurs remains unclear. Building on theories of counterfactual thinking we argue the ECE is more likely to occur when people experience counterfactual thought or emotion and that a key trigger is a negative versus positive task outcome. Participants either selected a drink (Experiment 1) or chocolate (Experiment 2) from a limited (6) versus extensive (24) selection (Experiment 1) or were given no choice versus extensive (24) choice (Experiment 2). In both experiments, however, the choice was illusory: Half the participants tasted a 'good' flavour, half a 'bad' flavour. As predicted, extensive choice was only detrimental to satisfaction when participants tasted the 'bad' drink or chocolate, and this was mediated by the experience of counterfactual thought (Experiment 1) or emotion (Experiment 2). When outcomes were positive, participants were similarly satisfied with limited versus extensive and no choice versus extensive choice. Implications for our theoretical understanding of the ECE and for the construction of choice architectures aimed at promoting individual satisfaction and well-being are discussed.16 page(s