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1	Accuracy of Computer Generated Approximations to Julia Sets Hoggard, John W. 17 August 2000 (has links) A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed. / Ph. D. tangent meromorphic computer algorithms polynomial Schwarzian derivative Julia sets
2	[pt] COMPORTAMENTO ESTATÍSTICO DE PRODUTOS TORTOS: DERIVADA SCHWARZIANA E LEIS DO ARCO-SENO / [en] STATISTICAL BEHAVIOR OF SKEW PRODUCTS: SCHWARZIAN DERIVATIVE AND ARC-SINE LAWS RAUL STEVEN RODRIGUEZ CHAVEZ 11 June 2024 (has links) [pt] Consideramos produtos tortos sobre shifts de Bernoulli, cuja dinâmica fibrada é dada por difeomorfismos do intervalo. Estudamos o comportamento previsível e/ou histórico destes sistemas, referindo-nos à convergência e/ou não convergência, da média de Birkhoff, respectivamente. Utilizamos a derivada Schwarziana das fibras e a lei do arco-seno para identificar condições nas quais esses produtos tortos apresentam esses tipos de comportamento. Identificamos distintos tipos de comportamento em relação à derivada Schwarziana. Quando a derivada Schwarziana é negativa, o produto torto tem bacias entrelaçadas. Por outro lado, quando a derivada Schwarziana é positiva, o produto torto possui uma medida física. Finalmente, quando a derivada Schwarziana é nula, o produto torto tem comportamento histórico. No último cenário, estabelecemos uma conexão entre o comportamento histórico e a lei do arco-seno que nos permite obter resultados em outras configurações independentes do sinal da derivada Schwarziana. / [en] We consider skew products over Bernoulli shifts, whose fibred dynamics is given by diffeomorphisms of the interval. We study the predictable and/or historical behavior, referring to convergence and/or non-convergence, of the Birkhoff average, respectively. We employ the Schwarzian derivative of the fiber maps and the arc-sine law to identify conditions under which these skew products exhibit these types of behavior. We identify distinct types of behavior according to the Schwarzian derivative. When the Schwarzian derivative is negative, the skew product has intermingled basins. Conversely, when the Schwarzian derivative is positive, the skew product has a physical measure. Finally, when the Schwarzian derivative is zero, the skew product has historical behavior. In the latter scenario, we establish a connection between historical behavior and the arc-sine law that allows us to obtain results in other settings independent of the sign of the Schwarzian derivative. [pt] BACIAS ENTRELACADAS [pt] LEI DO ARCO-SENO [pt] DERIVADA SCHWARZIANA [pt] COMPORTAMENTO HISTORICO [pt] MEDIDA FISICA [en] INTERMINGLED BASINS [en] ARC-SINE LAW [en] SCHWARZIAN DERIVATIVE [en] HISTORICAL BEHAVIOR [en] PHYSICAL MEASURE
3	On multifractality, Schwarzian derivative and asymptotic variance of whole-plane SLE / Sur la mutifractalité, la dérivée schwarziene et la variance asymptotique de whole-plane SLE Ho, Xuan Hieu 05 December 2016 (has links) Soit f une instance du whole-plane $\SLE_\kappa$ : on sait que pour certaines valeurs de κ, p les moments dérivés $\mathbb{E}(\vert f'(z) \vert^p)$ peuvent être écrits sous une forme fermée, étude qui a permis de mettre au jour une nouvelle phase du spectre des moyennes intégrales. Le but de cette thèse est une étude des moments généralisés $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : cette étude permet de confirmer la structure algébrique riche du whole-plane SLE. On montre que les formes fermées des moments mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ apparaissent sur une famille dénombrable de paraboles du plan (p, q), en étendant les équations de Beliaev-Smirnov à ce cas. Nous introduisons également le spectre généralisé β(p, q; κ), correspondant au comportement asymptotiques des moyennes intégrales mixtes. Le spectre généralisé moyen du whole-plane SLE prend quatre formes possibles, séparés par cinq séparatrices dans $\R^2$. Nous proposons également une approche semblable pour la dérivée Schwarziene S(f)(z) de l’application de SLE. Les calculs sur les équations de Beliaev-Smirnov d’une certaine générale forme de moment mène à une formulation explicite de $\mathbb{E}(S(f)(z))$ . Nous étudions finalement la variance asymptotique de McMullen et démontrons une relation entre la croissance infinitésimale du spectre de la moyenne intégrale et la variance asymptotique pour SLE₂. / Let f an instance of the whole-plane $\SLE_\kappa$ conformal map from the unit disk D to the slit plane: We know that for certain values of κ, p the derivative moments $\mathbb{E}(\vert f'(z) \vert^p)$ can be written in a closed form, study that has updated a new phase of the integral means spectrum. The goal of this thesis is a study on generalized moments $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : ΒββThis study permit confirm the rich algebraic structure of the whole-plane version of SLE. It will be showed that closed forms of the mixed moments E mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ can be obtained on a countable family of parabolas in the moment plane (p, q), by extending the so-called Beliaev–Smirnov equation to this case. We also introduce the generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the mixed moments. The average generalized spectrum of whole-plane SLE takes four possible forms, separated by five phase transition lines in $\R^2$. We also propose a similar approach for the Schwarzian derivative S(f)(z) of SLE maps. Computations on the Beliaev–Smirnov equation of a certain general form of moment lead to an explicit formula of $\mathbb{E}(S(f)(z))$ . We finally study the McMullen asymptotic variance and prove a relation between the infinitesimal growth of the integral mean spectrum and the asymptotic variance in an expectation sense for SLE₂. Whole-plane SLE Moment logarithmique Equation de Beliaev-Smirnov Spectre de la moyenne intégrale Dérivée Schwarziene Variance asymptotique de McMullen Whole-plane SLE Logarithmic moment Beliaev-Smirnov equation Generalized integral means spectrum Schwarzian derivative McMullen’s asymptotic variance 514.742
4	Feigenbaum Scaling Sendrowski, Janek January 2020 (has links) In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive. Feigenbaum Feigenbaum constant Feigenbaum point superstable self-similarity fractals final-state diagram logistic map period-doubling operator linearized period-doubling operator Cantor set Sharkovsky sequence Chebyshev interpolation stable manifold unstable manifold orbits fixed points scaling factor pitchfork bifurcation cobweb plot renormalization iterates one-hump map unimodal map bifurcation cascade Feigenbaum universality chaos theory spectrum generalized Feigenvalues bifurcation points superstable points fixed function universal function eigenvalues eigenvectors eigenfunctions universality conditions Schwarzian derivative Newton’s method power method Arnoldi’s method attractive fixed point repellent fixed point Banach fixed-point theorem collocation method functional analysis topology Mathematical Analysis Matematisk analys Other Mathematics Annan matematik Computational Mathematics Beräkningsmatematik

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