Dynamics of Modular Matings

In the paper 'Mating quadratic maps with the modular group II' the current authors proved that each member of the family of holomorphic $(2:2)$ correspondences $\mathcal{F}_a$: $$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right) +\left(\frac{aw-1}{w-1}\right)^2=3,$$ introduced by the first author and C. Penrose in 'Mating quadratic maps with the modular group', is a mating between the modular group and a member of the parabolic family of quadratic rational maps $P_A:z\to z+1/z+A$ whenever the limit set of $\mathcal{F}_a$ is connected. Here we provide a dynamical description for the correspondences $\mathcal{F}_a$ which parallels the Douady and Hubbard description for quadratic polynomials. We define a B\"ottcher map and a Green's function for $\mathcal{F}_a$, and we show how in this setting periodic geodesics play the role played by external rays for quadratic polynomials. Finally, we prove a Yoccoz inequality which implies that for the parameter $a$ to be in the connectedness locus $M_{\Gamma}$ of the family $\mathcal{F}_a$, the value of the log-multiplier of an alpha fixed point which has combinatorial rotation number $1/q$ lies in a strip whose width goes to zero at rate proportional to $(\log q)/q^2$

Parabolic-like maps

In this paper we introduce the notion of parabolic-like mapping, which is an object similar to a polynomial-like mapping, but with a parabolic external class, i.e. an external map with a parabolic fixed point. We prove a straightening theorem for parabolic-like maps, which states that any parabolic-like map of degree 2 is hybrid conjugate to a member of the family Per_1(1), and this member is unique (up to holomorphic conjugacy) if the filled Julia set of the parabolic-like map is connected.Comment: 32 pages, 12 figure

Blendas de bagaço de cana-de-açúcar, podas de mangueira e cajueiro: caracterização das propriedades e investigação de seus potenciais energéticos

O trabalho apresenta o potencial energético de blendas de bagaço de cana-de-açúcar e podas de mangueira ecajueiro. As amostras foram caracterizadas por meio da determinação do poder calorífico superior, densidade,análise imediata (umidade, cinzas, matéria volátil e carbono fixo), análise elementar e termogravimetria. Asmisturas foram preparadas com diferentes frações em massa. Após avaliação qualitativa e quantitativa dasamostras verificou-se quais blendas apresentaram os melhores resultados e, em seguida, os briquetes foramproduzidos. As misturas de bagaço de cana:poda de mangueira (frações 50%: 50%), bagaço de cana:poda decajueiro (frações 50%: 50%) e bagaço de cana:poda de cajueiro:poda de mangueira (frações 50%: 25%: 25%)foram utilizadas na produção dos briquetes. A última parte do trabalho apresenta a diferença entre o potencialenergético dos briquetes e do bagaço de cana. A densidade energética dos briquetes de bagaço de cana-deaçúcare poda de mangueira aumentou de 1,79 para 13,55 (kJ cm-3), em relação ao material não compactado.Os briquetes com poda de cajueiro variaram de 1,97 a 12,75 (kJ cm-3) e os briquetes de bagaço de cana:podade cajueiro:poda de mangueira variaram de 2,46 a 12,18 (kJ cm-3). O briquete de bagaço de cana apresentouvariação de 1,25 para 13,22 (kJ cm-3). A investigação do potencial energético de bioprodutos agrícolas (podasde mangueira e cajueiro) em mistura com bagaço de cana-de-açúcar, apresenta à indústria sucroalcooleirauma nova possibilidade de reutilização de resíduos de biomassa para obtenção de energia.Palavras-chave: Sustentabilidade; Biomassa; Briquetes; Poda de mangueira; Poda de cajueiro

Parabolic-like mappings and correspondences

Non UBCUnreviewedAuthor affiliation: University of Sao PauloPostdoctora

On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

In the paper 'On the dynamics of polynomial-like mappings' Douady and Hubbard introduced the notion of polynomial-like maps. They used it to identify homeomophic copies of the Mandelbrot set inside the Mandelbrot set. They conjectured that in case of primitive copies the homeomorphism between the homeomorphic copy of the Mandelbrot set and the Mandelbrot set is q.-c., and similarly in the satellite case, it is q.-c. off any small neighborhood of the root. These conjectures are now Theorems due to Lyubich. The satellite copies of the Mandelbrot set are clearly not q-c homeomorphic to the Mandelbrot set. But are they mutually q-c homeomorphic? Or even q-c homeomorphic to half of the logistic Mandelbrot set? In this paper we prove that, in general, the induced Douady-Hubbard homeomorphism is not the restriction of a q-c homeomorphism: For any two satellite copies of the Mandelbrot set, the induced Douady-Hubbard homeomorphism is not q-c, if the root multipliers, which are primitive q and q' roots of unity, have q different from q'

On parabolic external maps

We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings