Geometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n → ∞

We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle

Complex perspective for the projective heat map acting on pentagons

We place Schwartz's work on the real dynamics of the projective heat map H into the complex perspective by computing its rst dynamical degree and gleaning some corollaries about the dynamics of H

Superstable manifolds of invariant circles and codimension-one Böttcher functions

Let f : X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n>1. Suppose that there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose that f restricted to this line is given by z↦zb, with resulting invariant circle S. We prove that if a≥b, then the local stable manifold Wsloc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a≥b cannot be relaxed without adding additional hypotheses by presenting two examples with a<b for which Wsloc(S) is not real analytic in the neighborhood of any point

Rational Map of CP^2 with No Invariant Foliation

Conference Poster presented at: Midwest Dynamical Systems Conference, Champaign/Urbana, IL November 1-3, 2013

Rational Maps of CP^2 with No Invariant Foliation

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation

Results and Examples Regarding Bifurcation with a Two-Dimensional Kernal

Many problems in pure and applied mathematics entail studying the structure of solutions to F(x; y) = 0, where F is a nonlinear operator between Banach spaces and y is a real parameter. A parameter value where the structure of solutions of F changes is called a bifurcation point. The particular method of analysis for bifurcation depends on the dimension of the kernel of DxF(0,λ), the linearization of F. The purpose of our study was to examine some consequences of a recent theorem on bifurcations with 2-dimensional kernels. This resent theorem was compared to previous methods. Also, some specific classes of equations were identified in which the theorem always holds, and an algebraic example was found that illustrates bifurcations with a 2-dimensional kernel

Development of a simplified ray path model for estimating the range and depth of vocalising marine mammals

A simplified ray path model has been developed to simulate various source, receiver geometries. The difference in the arrival time of the multi-path signals (surface and seabed reflections) were calculated and compared with those measured on recorded data obtained during sea trials. A number of assumption have been made in initial models including a constant sound velocity-depth profile and the treatment of the surface and seabed as a simple reflecting surfaces. Initial results have shown a number of examples with a reasonable correlation between estimated position of a submerged cetacean and the associated surface observations. Examples of multiple (positioning) solutions were however found, these are in the main thought to be due to imprecision in the knowledge of the hydrophone and water depth and inaccuracies in the initial timing measurements. The use of correlation techniques and stand-alone depth measurement devices is therefore proposed for future measurements and analysis using this technique. It is felt that within constraints, this technique provides valuable additional information regarding cetacean behaviour in the wild and can be used on recorded data sets to validate observer records. The addition of more complex time measurement techniques and better ray path modelling will hopefully provide a useful analysis tool in the study of cetaceans

Trophic interactions will expand geographically but be less intense as oceans warm

Interactions among species are likely to change geographically due to climate-driven species range shifts and in intensity due to physiological responses to increasing temperatures. Marine ectotherms experience temperatures closer to their upper thermal limits due to the paucity of temporary thermal refugia compared to those available to terrestrial organisms. Thermal limits of marine ectotherms also vary among species and trophic levels, making their trophic interactions more prone to changes as oceans warm. We assessed how temperature affects reef fish trophic interactions in the Western Atlantic and modeled projections of changes in fish occurrence, biomass, and feeding intensity across latitudes due to climate change. Under ocean warming, tropical reefs will experience diminished trophic interactions, particularly herbivory and invertivory, potentially reinforcing algal dominance in this region. Tropicalization events are more likely to occur in the northern hemisphere, where feeding by tropical herbivores is predicted to expand from the northern Caribbean to extratropical reefs. Conversely, feeding by omnivores is predicted to decrease in this area with minor increases in the Caribbean and southern Brazil. Feeding by invertivores declines across all latitudes in future predictions, jeopardizing a critical trophic link. Most changes are predicted to occur by 2050 and can significantly affect ecosystem functioning, causing dominance shifts and the rise of novel ecosystems.Postprint6,86