Surface Acoustic Waves on Semiconductors

The paper is concerned with a discussion of SAW properties on semiconductor substrates. Some applications are briefly highlighted.Zadanie pt. Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki nr 885/P-DUN/2014 zostało dofinansowane ze środków MNiSW w ramach działalności upowszechniającej naukę

Virtual Immediate Basins of Newton Maps and Asymptotic Values

Newton's root finding method applied to a (transcendental) entire function f:C->C is the iteration of a meromorphic function N. It is well known that if for some starting value z, Newton's method converges to a point x in C, then f has a root at x. We show that in many cases, if an orbit converges to infinity for Newton's method, then f has a `virtual root' at infinity. More precisely, we show that if N has an invariant Baker domain that satisfies some mild assumptions, then 0 is an asymptotic value for f. Conversely, we show that if f has an asymptotic value of logarithmic type at 0, then the singularity over 0 is contained in an invariant Baker domain of N, which we call a virtual immediate basin. We show by way of counterexamples that this is not true for more general types of singularities.Comment: 15 pages, 1 figur

Scaling Ratios and Triangles in Siegel Disks

Let $f(z)=e^{2i\pi\theta} z+z^2$, where $\theta$ is a quadratic irrational. McMullen proved that the Siegel disk for $f$ is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if $\theta=(\sqrt 5-1)/2$ is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 year old conjecture.Comment: 13 pages, 13 PostScript figure

From local to global analytic conjugacies

Let $f_1$ and $f_2$ be rational maps with Julia sets $J_1$ and $J_2$, and let $\Psi:J_1\to \mathbb{P}^1$ be any continuous map such that $\Psi\circ f_1=f_2\circ \Psi$ on $J_1$. We show that if $\Psi$ is $\mathbb{C}$-differentiable, with non-vanishing derivative, at some repelling periodic point $z_1\in J_1$, then $\Psi$ admits an analytic extension to $\mathbb{P}^1\setminus {\mathcal E}_1$, where ${\mathcal E}_1$ is the exceptional set of $f_1$. Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if ${\mathcal E}_1=\emptyset$ then the extended map $\Psi$ is rational, and in this situation $\Psi(J_1)=J_2$ and $\Psi^{-1}(J_2)=J_1$, provided that $\Psi$ is not constant. On the other hand, if ${\mathcal E}_1\neq \emptyset$ then the extended map may be transcendental: for example, when $f_1$ is a power map (conjugate to $z\mapsto z^{\pm d}$) or a Chebyshev map (conjugate to \pm \text{Х}_d with \text{Х}_d(z+z^{-1}) = z^d+z^{-d}), and when $f_2$ is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof