The renormalization theory of critical circle maps was developed in the late 1970's-early 1980's to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps. In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation.The first renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the setting of unimodal maps. The recent spectacular progress in the unimodal renormalization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS]. He introduced methods of holomorphic dynamics and Teichmüller theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renormalizations of unimodal maps of bounded combinatorial type converge to a horseshoe attractor. Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate. And finally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion.The renormalization theory of circle maps has developed alongside the unimodal theory. The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type. Later de Faria and de Melo [dFdM2] used McMullen's work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound.Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case. The first part of Lyubich's work was to to endow the This work was partially supported by NSF Grant DMS-9804606.2 M I C H A E L Y A M P O L S K Y ambient space of the renormalization transformation with the structure of a complexanalytic manifold, with respect to which renormalization is an analytic operator. He then showed that the stable sets of periodic points of this operator are codimension one submanifolds and used an argument based on McMullen's Tower Rigidity Theorem and an infinite dimensional version of the Schwarz Lemma to show that renormalization is a strict contraction in the stable direction. The second part of Lyubich's work was to show the existence of an unstab...

Background-Placentas are generally round-oval in shape, but "irregular" shapes are common. In the Collaborative Perinatal Project data, irregular shapes were associated with lower birth weight for placental weight, suggesting variably shaped placentas have altered function.

Goal: We assess the effect on placental efficiency of the non-centrality of the umbilical cord insertion and on chorionic vascular distribution to determine if cord centrality measurably affects placental function as reflected in birth weight. Materials and Methods:1,225 placentas collected from a prospective cohort had digital photographs of the chorionic plate. Of these, 1023 were term, 44 had velamentous cord insertion and 12 had missing clinical data, leaving N=967 (94.5 %) cases for analysis. Mathematical tools included a dynamical stochastic growth model of placental vasculature, Fourier analysis of radial parameterization of placental perimeters, and relative chorionic vascular density (a measure of "gaps" in the vascular coverage) derived from manual tracings of the fetal chorionic surface images. Bivariate correlations used Pearson's or Spearman's rank correlation as appropriate, with p<0.05 considered significant. Results:The correlation of the standard deviation of the placental radius (a measure of nonroundness of the placenta) with cord displacement was negligible (r=0.01). Empirical simulations of the vascular growth model with cord displacement showed no deviation from a normal round-tooval placental shape for cord displacement of 10 -50% of placental radius. The correlation of the metabolic scaling exponent β with cord displacement measured by Fourier analysis is 0.17 (p < 0.001). Analysis of the chorionic vascular density in traced images shows a high correlation of the © 2009 Elsevier Ltd. All rights reserved.Corresponding author: Michael Yampolsky, PhD Department of Mathematics University of Toronto 40 St. George Street, Toronto, Canada M5S2E4 yampolsky.michael@gmail.com. Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. NIH Public Access Author ManuscriptPlacenta. Author manuscript; available in PMC 2010 December 1. Conclusion:Non-central cord insertion has little measurable correlation with placental shape in observed or simulated placentas. However, placentas with a displaced cord show a markedly reduced transport efficiency, reflected in a larger value of β and hence in a smaller birth weight for a given placental weight. Placentas with a non-central cord insertion have a sparser chorionic vascular distribution, as measured by the relative vascular distance. Even if typically a placenta with a noncentral insertion is of a normal round shape, its vasculature is less metabolically effective. These findings demonstrate another method by which altered placental structure may affect the fetal environment, influencing birth weight and potentially c...

Goal In clinical practice, variability of placental surface shape is common. We measure the average placental shape in a birth cohort and the effect deviations from the average have on placental functional efficiency. We test whether altered placental shape improves the specificity of histopathology diagnoses of maternal uteroplacental and feto-placental vascular pathology for clinical outcomes. Materials and Methods 1225 placentas from a prospective cohort had chorionic plate digital photographs with perimeters marked at 1–2 cm intervals. After exclusions of preterm (n=202) and velamentous cord insertion (n=44), 979 (95.7%) placentas were analyzed. Median shape and mean perimeter were estimated. The relationship of fetal and placental weight was used as an index of placental efficiency termed “β”. The principal placental histopathology diagnoses of maternal uteroplacental and fetoplacental vascular pathologies were coded by review of individual lesion scores. Acute fetal inflammation was scored as a“negative control” pathology not expected to affect shape. ANOVA with Bonferroni tests for subgroup comparisons were used. Results The mean placental chorionic shape at term was round with a radius estimated at 9.1 cm. Increased variability of the placental shape was associated with lower placental functional efficiency. After stratifying on placental shape, the presence of either maternal uteroplacental or fetoplacental vascular pathology was significantly associated with lower placental efficiency only when shape was abnormal. Conclusions Quantifying abnormality of placental shape is a meaningful clinical tool. Abnormal shapes are associated with reduced placental efficiency. We hypothesize that such shapes reflect deformations of placental vascular architecture, and that an abnormal placental shape serves as a marker of maternal uteroplacental and/or fetoplacental vascular pathology of sufficiently long standing to impact placental (and by extension, potentially fetal) development.

Background We tested the hypothesis that the fetal-placental relationship scales allometrically and identified modifying factors of that relationship. Materials and Methods Among women delivering after 34 weeks but prior to 43 weeks gestation, 24,601 participants in the Collaborative Perinatal Project (CPP) had complete data for placental gross proportion measures, specifically, placental weight (PW), disk shape, larger and smaller disk diameters and thickness, and umbilical cord length. The allometric metabolic equation was solved for α and β by rewriting PW = α (BW)^β as ln (PW) = lnα + β [ln(BW)]. αɩ was then the dependent variable in regressions with p<0.05 significant. Results Mean β was 0.78± 0.02 (range 0.66, 0.89), which is consistent with the scaling exponent 0.75 predicted by Kleiber's Law.. Gestational age, maternal age, maternal BMI, parity, smoking, socioeconomic status, infant sex, and changes in placental proportions each had independent and significant effects on α. Conclusions We find an allometric scaling relation between the placental weight and the birth weight in the CPP cohort with an exponent approximately equal to 0.75, as predicted by Kleiber's Law. This implies that: 1) Placental weight is a justifiable proxy for fetal metabolic rate when other measures of fetal metabolic rate are not available; and 2) The allometric relationship between placental and birth weight is consistent with the hypothesis that the fetal-placental unit functions as a fractal supply limited system. Furthermore, our data suggest that the maternal and fetal variables we examined have at least part of their effects on the normal balance between placental weight and birth weight via effects on gross placental growth dimensions.