Laplacians on a family of quadratic Julia sets. Part I (Q2841339)

Analysis on fractals is a relatively young subject and since it conception in the 1980s it has been widely developed, see for instance [\textit{B. Branner}, The Mandelbrot set, Chaos and fractals. (Providence, RI, 1988) Proc. Sympos. Appl. Math., Vol. 39, Amer. Math. Soc., Providence pp. 75-105 (1989)], [\textit{J. Kigami}, Analysis on fractals. Cambridge Tracts in Mathematics. 143. Cambridge: Cambridge University Press (2001; Zbl 0998.28004)], [\textit{R. S. Strichartz}, Differential equations on fractals. A tutorial. Princeton, NJ: Princeton University Press (2006; Zbl 1190.35001)] and references therein. The approach pioneered by Kigami [loc. cit.] is to construct a Laplacian from energy forms and measures. Many of the examples which have been studied thus far have been strictly self-similar sets that arise as fixed points of iterated function systems, such as the Sierpinski gasket, Sierpinski carpet, hexaflake and Vicsek snowflake. However, \textit{L. G. Rogers} and \textit{A. Teplyaev} [Commun. Pure Appl. Anal. 9, No. 1, 211--231 (2010; Zbl 1194.28013)] have extended Kigami's approach to the Basilica Julia set. In the paper under review, in the same spirit, the authors show the existence of Laplacians on Julia sets \(J_{c}\) resulting from hyperbolic quadratic polynomials, namely polynomials of the form \(P_{c}(z) = z^{2} + c\) where \(c\) belongs to the Mandelbrot set. This class of Julia sets include both the Basilica and the Douady rabbit. (Note that the Laplacian on the Basilica given in [Zbl 1194.28013] results from a different energy form than that considered in the article under review). The Julia sets \(J_{c}\) are a well studied class of nonlinear fractals that are both connected and finitely ramified, meaning that they may be disconnected by removing a finite number of points. The major difficulty, which is overcome in the current work, comes from the nonlinearity of \(J_{c}\), specifically in that one does not have a strict self-similar structure.NEWLINENEWLINEStarting from graph approximations \(V_{m}\) of the Julia set \(J_{c}\) provided by the dynamics, the authors first define an energy forms \(\mathcal{E}_{m}^{(j)}\) dependent on the geometry of \(V_{m}\), for \(j \in \{ 0, 1, \dots, k \}\), where \(k\) denotes the number of points in \(V_{m}\) that are identified to obtain a single point in \(V_{m+1} \setminus V_{m}\). They then show that the energy form \(\mathcal{E}^{(j)}(\cdot, \cdot) \mathrel{:=} \lim_{m \to \infty} \mathcal{E}^{(j)}_{m}(\cdot, \cdot)\) is well defined and finally define the energy form \(\mathcal{E}(\cdot, \cdot)\) by NEWLINE\[NEWLINE \mathcal{E}(\cdot, \cdot) \mathrel{:=} \frac{1}{k} \sum_{j = 0}^{k-1} \mathcal{E}^{(j)}(\cdot, \cdot). NEWLINE\]NEWLINE This energy form is then used to define a Laplacian by the weak formulation, once a measure has been fixed. There are two natural measures which are considered: the \(P_{c}\)-invariant measure \(\mu\) which ignores the geometric aspects and the conformal measure \(\nu\) which is not \(P_{c}\)-invariant but does transform according to a power of the Jacobian of the mapping \(P_{c}\). More specifically, the measure \(\mu\) is the pullback to \(J_{c}\) of the normalised Lebesgue measure on the circle approximated on \(V_{m}\) by the discrete measure that gives equal weight to all vertices and satisfies the \(P_{c}\)-invariance condition NEWLINE\[NEWLINE \int_{J_{c}} f \circ P_{c} \, \mathrm{d}\mu = \int_{J_{c}} f \mathrm{d}\mu. NEWLINE\]NEWLINE The conformal measure \(\nu\) is characterised by the condition NEWLINE\[NEWLINE \int_{J_{c}} f \circ P_{c} | P_{c}' |^{\delta} \, \mathrm{d}\mu = \int_{J_{c}} f \mathrm{d}\mu, NEWLINE\]NEWLINE for a constant \(\delta\) that may be identified as the Hausdorff dimension of \(J_{c}\).NEWLINENEWLINEIt is shown that the Laplacian \(\Delta_{\mu}\) constructed using the measure \(\mu\) satisfies a \(P_{c}\)-invariance condition NEWLINE\[NEWLINE \Delta_{\mu}(u \circ P_{c}) = 2^{1+1/k}(\Delta_{\mu}(u)) \circ P_{c}. NEWLINE\]NEWLINE In particular, if \(u\) is a \(\lambda\)-eigenfunction, then \(u \circ P_{c}\) is a \(2^{1+1/k} \lambda\)-eigenfunction. In other words, the spectrum of \(\Delta_{\mu}\) is preserved under multiplication by \(2^{1+1/k}\). Further, this Laplacian depends only on the topological type of \(J_{c}\). Thus, for quasicircles, the operator \(\Delta_{\mu}\) is the usual Laplacian on the unit circle with the parameterisation of a quasiconformal map. In comparison, the conformal Laplacian \(\Delta_{\nu}\) is built from the measure \(\nu\) and depends on the geometry of \(J_{c}\).NEWLINENEWLINEThe authors also describe numerical procedures to approximate the eigenvalues and eigenfunctions of the Laplacians \(\Delta_{\mu}\) and \(\Delta_{\nu}\) and present the computational results. They give a classification of the eigenfunctions and present a detailed explanation of the computational data. Further, motivated by this data, the authors present conjectures on the eigenvalue counting functions of these Laplacians.NEWLINENEWLINEThe work of the article under review is continued in [\textit{T. Aougab} et al., Commun. Pure Appl. Anal. 12, No. 1, 1--58 (2013; Zbl 1264.28003)]. In brief, here the authors show that the methods of the current article can be applied to a larger class of Julia sets and give a systematic method that reduces the construction of an invariant energy to the solution of a nonlinear finite-dimensional eigenvalue problem.