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$B$3$l$^$G$N(B Kyoto Dynamics Days
In this lecture we will describe a number of different ways that Sierpinski curves (that is, sets homeomorphic to the Sierpinski carpet fractal) arise as the Julia sets of complex rational functions. While all these Julia sets are the same from a topological point of view, they are all very different from a dynamical systems point of view. We'll explain why.
A polynomial skew product on C^2 is a regular polynomial map of degree at least two such that the first component depends only on the first coordinate. The second Julia set of a polynomial skew product might have symmetries, that is, it might be invariant under some linear maps on C^2. We investigate the structure of the group of symmetries. We show that, except for two types, polynomial skew products having the same second Julia set are essentially the same.
Informally speaking, a compact set in the plane is computable, if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. In a joint work with M. Braverman, we consider the question of computability of Julia sets of quadratic polynomials. I will discuss some of the surprising results of our study.
We establish a one-to-one correspondence between the renormalizations and proper totally invariant closed sets (i.e., alpha-limit sets) of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. We describe the minimal renormalization by constructing the minimal totally invariant closed set, so that we can define the renormalization operator. Using consecutive renormalizations, we obtain complete topological characterization of alpha-limit sets and nonwandering set decomposition. For piecewise linear Lorenz map with slopes no less than 1, we show that each renormalization is periodic and every proper alpha-limit set is countable.
The paper can be obtained via math.DS/0703777.
wild $B$J(Bhomoclinic class $B$r@]F0$9$k$3$H$G!"(B universal dynamics $B$H$$$&J#;($J9=B$$r;}$C$?NO3X7O$r9=@.$9$k$3$H$,$G$-$k!#(B
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Doyle & McMullen $B$NO@J8(B$B$K$*$$$F!"%j!<%^%s5eLL>e$NB?LLBN72$N:nMQ$H2D49$H$J$k@5B'
S.Crass; A family of critically finite maps with symmetry, Publ. Mat. 49(1), 127-157, 2005$B!!(Bmath.DS/0307057
P.Doyle & C.McMullen; Solving the quintic by iteration, Acta Mathematica 163, 80-151, 1989
K.Maegawa; Holomorphic maps on P^k with sparse critical orbits, submitted
K.Ueno; Dynamics of symmetric holomorphic maps on projective spaces, Publ. Mat. 51(2), 333-344, 2007 arXiv:0707.3496
The goal of this talk is to describe the dynamics of the local map of coupled map lattice(CML) -- the discrete versions of models. Following Kaneko I view CMLs as phenomenological models of the medium (which is assumed to be homogeneous and unbounded) and I present the dynamical system approach to the analysis of the global behavior of solutions of CML developed in works of V. Afraimovich, M. Brin, D. Orendovici, and Y. Pesin. This analysis is aimed at establishing spatio-temporal chaos associated with the set of traveling wave solutions of CML and describing the dynamics of the evolution operator on this set. The main results claim that the dynamics of the evolution operator on the set of traveling wave solutions is completely determined by the dynamics of the local map thus making the study of the latter as the primary goal of this work. In this talk, we see how CMLs are related to traveling waves and transited to PDEs, review various types of CMLs and local maps, and then, study their dynamics -- hyperbolic and ergodic properties.
By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that the Julia set of a polynomial is a Cantor set if and only if each component of the filled-in Julia set containing critical points is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992.
$BNO3X7O!&B?JQ?tH!?tO@9gF1%;%_%J!<(BFrancois Berteloot (Toulouse)
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Part 1: A unified approach
(speaker: Marco Abate, Universita' di Pisa)
Part 2: The case of holomorphic maps
(speaker: Francesca Tovena, Universita' di Roma Tor Vergata)
Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete-symmetries and can form in a random environment. In this talk, I will describe an N-particle based Hamiltonian theory which, if formulated in terms of the interparticle distances, leads to the analysis of a non-normal `configuration' matrix whose nullspace structure determines the existence or non-existence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns, in principle, can be identified and calculated by a random- walk scheme which systematically uses the m-smallest singular values as a ratchet mechanism to hone in on lattices with many new properties, including a complete lack of discrete symmetries and heterogeneous particle strengths.
I will discuss the thermodynamical formalism for dynamical systems admitting inducing schemes (tower constructions). This includes some real unimodal and multimodal maps, nonuniformly expanding maps, Henon attractors, etc. I discuss existence, and uniqueness of equilibrium (Gibbs) measures for a broad class of potential functions including absolutely continuous (SRB) measures and measures of maximal entropy.
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