% talk in Urbana-Champaign, 2003 \documentclass{slides} \usepackage{amssymb,amsmath,exscale,latexsym,amsthm,graphics} \usepackage{epsfig} % eps-Bilder einlesen \usepackage{epic} % irgendsone Graphikerweiterung \usepackage{eepic} % nochne erweiterung \usepackage{graphics} \usepackage[usenames]{color} \pagestyle{empty} % selbstdefinierte theoremumgebungen \theoremstyle{plain} \newtheorem*{theorem}{Theorem} \newtheorem{lemma}{Lemma} % \newtheorem{cor}[theorem]{Corollary} \theoremstyle{remark} \newtheorem*{remark}{Remark} \theoremstyle{remark} \newtheorem*{notation}{Notation} \theoremstyle{remark} \newtheorem*{acknowledgments}{Acknowledgments} % Hier wird das File mit meinen eigenen Definitionen geladen \input{definitionen} \begin{document} \begin{slide}% title \begin{center} {\Large Melting Snowballs} Daniel Meyer \\University of Washington \\ \vspace{3cm} \scalebox{1}{\includegraphics{snowball.eps}} %\caption{Embedding of the Snowball} \end{center} \end{slide} \begin{slide} A homeomorphism $f\colon X\to Y$ of metric spaces is called \emph{quasiconformal} if \begin{equation*} \varlimsup_{\epsilon\to0}\frac{\displaystyle{\max_{|x-a|=\epsilon}}|f(x)-f(a)|}{\displaystyle{\min_{|x-b|=\epsilon}}|f(x)-f(b)|}\leq K, \end{equation*} for all $x,a,b\in X$. It is called \emph{quasisymmetric} if there is a homeo- morphism $\eta\colon[0,\infty)\to[0,\infty)$, such that \begin{equation*} \frac{|x-a|}{|x-b|}\leq t \Rightarrow \frac{|f(x)-f(a)|}{|f(x)-f(b)|}\leq\eta(t), \end{equation*} for all $x,a,$ and $b$. \begin{center} \scalebox{0.8}{\includegraphics{elipse.eps}} \end{center} Quasisymmetry $\Rightarrow$ quasiconformality. \\ For $f\colon\R^n\to\R^n$ the two notions coincide. \\Coincide in a large class of spaces \\(Heinonen-Koskela '98). \end{slide} \begin{slide} Qc maps are best understood for $n=2$. We can assign the eccentricity and orientation of infinitesimal ellipses almost everywhere \begin{center} \scalebox{0.8}{\includegraphics{elipse.eps}} \end{center} by the \emph{measurable Riemann mapping Theorem} (Ahlfors-Bers '60). No analogue in higher dimensions. \end{slide} \begin{slide} There is a quasiconformal map $f\colon\R^2\to\R^2$, that maps the snowflake to the unit circle. \begin{center} \scalebox{1}{\includegraphics{snowflake.eps}} % \caption{Mapping of the Snowflake} \end{center} \emph{Quasicircles} (images of unit circle under a qc map $f\colon\R^2\to\R^2$) are characterized by the \emph{Ahlfors 3-point} condition. One can generate \emph{all} quasicircles by a snowflake type construction (Rohde '01). Higher dimensions ? \end{slide} \begin{slide}Problem: Embed ``fractal'' surface $S\subset \R^3$ quasisymmetrically in the plane. (David, Toro '99) $S$ with no rectifiable curves. (Cannon, Floyd, Parry '99) Combinatorial Riemann mapping theorem. (Bonk, Kleiner '02) Circle Packings. Kenyon (Bowers, Stephenson '97) Pentagon Subdivision . \end{slide} \begin{slide} \begin{center} Snowballs/Snowspheres \end{center} \begin{center} \scalebox{1}{\includegraphics{snowball.eps}} %\caption{Embedding of the Snowball} \end{center} Start with unit cube. Divide each face in $N\times N$ squares. Put small cubes \emph{symmetric} on some small squares. Divide each $\frac{1}{N}$ -face into $N\times N$ squares of side length $\frac{1}{N^2}$ in the same pattern. In the limit get the \emph{snowball} $\mathcal{B}$ and its \emph{snowsphere} $\mathcal{S}=\partial \mathcal{B}$. Works slightly more general. Different pattern and different $N_i\leq N_{\max } $ for each step. Could start with Tetrahedron. \smallskip \begin{remark} That $\frac{1}{N}$ -cubes fit into pyramid of height $\frac{1}{2}$ ensures that there are no self intersections. \end{remark} \end{slide} \begin{slide} %overlays \scalebox{1}{\includegraphics{pyramid.eps}} \scalebox{1.0}{\includegraphics{talk_pics/snowgen2.eps}} \scalebox{0.37}{\includegraphics{extend2.eps}} \end{slide} \begin{slide} \begin{theorem} Every snowsphere as above can be \newline mapped quasisymmetrically to the sphere $S^2$. \end{theorem} \textbf{Proof} will be illustrated for the self similar surface with the following \emph{generator}: \scalebox{1.0}{\includegraphics{talk_pics/snowgen.eps}} This has self intersections. Metric defined by $$d(x,y)=\inf \length \gamma,$$ where the infimum is taken over all curves $\gamma$ connecting $x$ and $y$ in $\mathcal{S}$ which do not cross through self intersections. If $\mathcal{S}$ is a surface in $\R^3$, $d(x,y)$ and $|x-y|$ are qs comparable. \end{slide} \begin{slide} Put conformal structure on generator. Charts: \scalebox{.9}{\includegraphics{talk_pics/2squares.eps}} \scalebox{.7}{\includegraphics{talk_pics/3quadrat.eps}} \scalebox{.7}{\includegraphics{talk_pics/5quadrat.eps}} By the uniformization theorem this is conformally equivalent to the disc. \end{slide} \begin{slide} The uniformized generator of the snowsphere: \begin{center} \scalebox{.57}{\includegraphics{talk_pics/snowtrace_color.eps}} \end{center} Each tile is conformally a square. \newline Two tiles which share a side may be mapped conformally to \raisebox{-.9ex}{\scalebox{.4}{\includegraphics{talk_pics/2squares.eps}}}. Copy this subdivision into each square by a conformal map. Keep subdividing conformally. This gives the embedding of squares and thus the embedding. \end{slide} \begin{slide} \scalebox{0.9}{\includegraphics{talk_pics/snowtrace.eps}} \end{slide} \begin{slide} \scalebox{0.9}{\includegraphics{talk_pics/snowtrace_small.eps}} \smallskip \scalebox{.9}{\includegraphics{talk_pics/2squares.eps}} \end{slide} \begin{slide} One can actually write down the conformal maps which realize the subdivision explicitly. Cut snowsphere along the diagonals into 4 pieces. Get again a self similar surface with generator: \begin{center} \scalebox{1.3}{\includegraphics{talk_pics/Rgen2.eps}} \end{center} Map this to the upper half plane, normalize by mapping the {\color{Red} left corner} to $-1$, the {\color{Green}right} to $1$ and the tip to $\infty$. The rational map is constructed by mapping white triangles to the upper half plane, {\color{Blue}blue triangles} to the lower half plane, normalize by black $\mapsto\infty$, {\color{Red}red} $\mapsto -1$ and {\color{Green}green} $\mapsto 1$. \end{slide} \begin{slide} From the construction one knows a lot about the rational map. The {\color{Red}red}, {\color{Green}green}, and black points are the critical points. The critical values $-1, 1$ and $\infty$ are fixpoints. The map is \emph{postcritically finite}. The rational map is \begin{align*} R(z)&=\lambda \frac{(z-1)(z-a_1)^3(z-a_2)^4(z-a_3)^5}{(z^2-{b_1}^2)^2(z^2-{b_2}^2)^2(z^2-{b_3}^2)^2}+1 \\ \label{Rsnow2} &=\mu \frac{(z+1)(z-c_1)^3(z-c_2)^4(z-c_3)^5}{(z^2-{b_1}^2)^2(z^2-{b_2}^2)^2(z^2-{b_3}^2)^2}-1, \end{align*} where $ a_1= -69.2485\dots, a_2= -0.726663\dots, \newline a_3= 3.33137\dots, b_1= 1.07729\dots, b_2= i \cdot 1.04067\dots, \newline b_3= i \cdot 18.7881\dots, \lambda = -0.00518147\dots\; .$ \bigskip \begin{theorem} Every self similar snowsphere may be realized by a rational map. \end{theorem} \end{slide} \begin{slide} Extension of the map. Extension of a qs map $f\colon\R^n\to\R^n$ to a qs map $f\colon\R^{n+1}\to\R^{n+1}$ was done by: $n=1$ Ahlfors and Beurling '56; $n=2$ Ahlfors '64; $n=3$ Carleson '74; any $n$ Tukia and V\"ais\"al\"a '82. \bigskip \begin{theorem} The embedding of the snowsphere can be extended to a qc map \begin{equation*} \bar{f}\colon\R^3\to\R^3. \end{equation*} \end{theorem} \end{slide} \begin{slide} Some ideas for the proof: \scalebox{0.37}{\includegraphics{extend.eps}} To each $n$-th approximation of the snowsphere $\mathcal{S}_n$ (in {\color{Blue} blue}), consider the surface \newline$T_n=\{x:\dist(x,\mathcal{S}_n)=c5^{-n}\}$ (in {\color{Red} red}). Decompose the shells bounded by $T_n$ and $T_{n+1}$ into simplices. Map these to shells bounded by spheres $r_nS^2$. The spheres $r_nS^2$ are uniformizations of $\mathcal{S}_n$. \end{slide} \begin{slide} Extension of qs maps: difficult. Extension of bi-Lipschitz maps : easy. For given cylinder consider hyperbolic triangle with same angles. \vspace{2cm} \begin{center} \scalebox{0.9}{\includegraphics{hyp_emb.eps}} \end{center} \vspace{2cm} By reflection this extends to a neighborhood. By Koebe the map is bi-Lipschitz on the hyperbolic triangle (up to scaling). Can extend this map to ``simplices''. \end{slide} \begin{slide} Dimensions and measures. Harmonic measure $=$ measure to integrate against to solve the Dirichlet problem. For simply connected $D\subset \C$ it is the image of Lebesgue measure on $\partial \D$ under the conformal map $f\colon \D\to D$. Let $D\subset\R^2$ be simply connected. For its harmonic measure $\omega_D$ we have \begin{equation*} \dim \omega_D = 1, \mbox{ Makarov '85}. \end{equation*} For every $n$ there is a number $\alpha(n) 2, \mbox{ Wolff '95}. \end{equation*} \end{slide} \begin{slide} Elliptic harmonic measure $\mu$ of a domain \newline $D\subset \R^n$ is the image of Lebesgue measure on $S^{n-1}$ under a quasiconformal map $f\colon \mathbb{B}\to D$. \smallskip \begin{theorem} The elliptic harmonic measure of the snowball $\mu_\mathcal{B}$ satisfies \begin{equation*} \dim \mu_\mathcal{B} < \dim \mathcal{S}. \end{equation*} \end{theorem} Conjectured : $\dim\mu_\mathcal{B}>2$. Numerically yes. \end{slide} \begin{slide} Let $\nu$ be invariant wrt. $R\colon\CDach\to \CDach$. \newline The \emph{Lyapunov exponent} is \begin{equation*} \chi_\nu:=\int\log|R'|\,d\nu. \end{equation*} The \emph{entropy} is \begin{equation*} h_\nu := \int\log J_\nu \,d\nu, \end{equation*} where $J_\nu=d\nu\comp R / d\nu$. Then \begin{equation*} \dim_\nu=\frac{h_\nu}{\chi_\nu} \end{equation*} Know $\dim\CDach=2$. \newline Lyapunov on Snowsphere $\chi(\mathcal{S})=\log 3$. \scalebox{0.8}{\includegraphics{mapdecomp.eps}} \end{slide} \begin{slide} Subdivisions, Origami with rational maps. \begin{center} \scalebox{0.8}{\includegraphics{subdivision.eps}} \end{center} This subdivision may be realized by the map $g(z)=1-\frac{2}{z^2}$. This is \emph{postcritically finite}. Thurston gave a characterization for such maps ('82, Douady-Hubbard '93). Is every pcf rational map realizable by a subdivision rule? Yes for polynomials (Poirier, '97). \end{slide} \begin{slide}%overlay \scalebox{.57}{\includegraphics{talk_pics/snowtrace_small.eps}} \end{slide} \begin{slide} Question: non-symmetric generators? \scalebox{.8}{\includegraphics{talk_pics/7x7.3squares.eps}} \end{slide} \end{document}