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Register a new userA geometric iterated function system on triangles
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We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.
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We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.
In a previous paper, dealing with Applications in $mathbb{R}^1$, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications in one dimension. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$. In our context, $L_s$ is studied in a space of $C^m$ functions and is not compact. Nevertheless, it is has a strictly positive $C^m$ eigenfunction $v_s$ with positive eigenvalue $lambda_s$ equal to the spectral radius of $L_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $lambda_s =1$. To compute the Hausdorff dimension of an IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we again approximate the eigenvalue problem by a collocation method, but now using continuous piecewise bilinear functions. Using the theory of positive linear operators and explicit a priori bounds on the partial derivatives of the strictly positive eigenfunction $v_s$, we are able to give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge to $s_*$ as the mesh size approaches zero. We also demonstrate by numerical computations that improved estimates can be obtained by the use of higher order piecewise tensor product polynomial approximations, although the present theory does not guarantee that these are strict upper and lower bounds. An important feature of our approach is that it also applies to the much more general problem of computing approximations to the spectral radius of positive transfer operators, which arise in many other applications.
Given a plane triangle $Delta$, one can construct a new triangle $Delta$ whose vertices are intersections of two cevian triples of $Delta$. We extend the family of operators $DeltamapstoDelta$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.
We discuss a version of Ecalles definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Omega-continuability, where Omega is a discrete filtered set, and show how to construct a universal Riemann surface X_Omega whose holomorphic functions are in one-to-one correspondence with Omega-continuable functions. We then discuss the Omega-continuability of convolution products and give estimates for iterated convolutions of the form hatphi_1*cdots *hatphi_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.
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