# 2006$BG/EY(B $B5~ET(B $BNO3X7O%;%_%J!<(B ## 2006 Kyoto Dynamical Systems seminar $BF|(B $B;~(B: $BKh=56bMK8a8e(B2$B;~$h$j(B $B>l(B $B=j(B: $B5~ETBg3XBg3X1!M}3X8&5f2J(B 1$B9f4[(B 3$B3,(B 314$B%;%_%J!<<<(B $B!J(B$B:rG/EY$H$O2q>l$,0[$J$j$^$9(B$B!K(B $B2a5n$N%;%_%J!<(B: 2005$BG/EY(B, 2004$BG/EY(B, 2003$BG/EY(B, 2002$BG/EY(B, 2001$BG/EY(B

$B$3$l$^$G$N(B Kyoto Dynamics Days

$B?7$7$$=gKJBYF"j^9!#(B • 1B7n(B12BF|!J6b!K(B B1|B<(B BMN2p!J5~ETBg3XBg3X1!?M4V!&4D6-3X8&5f2J!K(B B2DLsM-M} • 12B7n(B22BF|!J6b!K(B B>e86(B B?r?M!J6e=#Bg3XBg3X1!?tM}3XI\!K(B Chaotic dynamics in Painlev\'e VI • BNO3X7O!&B?JQ?tH!?tO@9gF1%;%_%J!<(B 12B7n(B12BF|!J2P!K8a8e(B1B;~(B30BJ,+i(B BM}3X8&5f2J(B 6B9f4[(B 2B3,(B 203B<<KF(B Jean-Yves Briend (Universite de Province, France) Quadratic polynomials over local fields • 12B7n(B8BF|!J6b!K(B Romain Dujardin (Universite de Paris 7) Continuity of Lyapunov exponents for polynomial automorphisms of C^2. Abstract: Motivated by a classical result in one variable complex dynamics, we prove that the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of \mathbb{C}^2 (for example: complex Henon mappings) vary continuously on parameter space. We also have a similar result for families degenerating to a one dimensional map. • BNO3X7O!&B?JQ?tH!?tO@9gF1%;%_%J!<(B 12B7n(B5BF|!J2P!K8a8e(B4B;~+i(B BM}3X8&5f2J(B 1B9f4[(B 5B3,(B 516B%;%_%J!<<<KF(B, 12B7n(B6BF|!J?e!K8aA0(B10B;~(B30BJ,+i(B BM}3X8&5f2J(B 1B9f4[(B 3B3,(B 321B%;%_%J!<<<K(B BF(B Romain Dujardin (Universite de Paris 7, France) Cubic polynomials: a measurable view on parameter space Abstract: The parameter space of cubic polynomials viewed as dynamical systems on \mathbb{C} is (a finite quotient of) \mathbb{C}^2. In these talks, I will describe how positive closed currents can be useful to describe the geography of this parameter space. The topics should include: Definition of the bifurcation currents. Distribution of critically preperiodic parameters. The bifurcation measure and Misiurewicz points. Deformations and laminar structure. Measurable similarity between dynamical and parameter space. Most of this is joint work with Charles Favre (CNRS). • 12B7n(B1BF|!J6b!K(B Christos Sourdis (B%"%F%MBg3X(B) Existence of heteroclinic orbits in a singularly perturbed system of O.D.E`s involving a corner layer • 11B7n(B24BF|!J6b!K(B B@iMU0o?M!J5~ETBg3XBg3X1!>pJs3X8&5f2J!K(B C^1 Approximation of Vector Fields on the Renormalization Group Method and its Applications B%"%V%9%H%i%/%H!'(B The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining solutions which are approximate to exact solutions uniformly in time. It is shown that for a given vector field on an arbitrary manifold, approximate solutions obtained on the RG method define a vector field which is close to an original vector field in C^1 topology. Furthermore, some topological properties of the approximate vector field, for instance, the existence of an invariant manifold and its stability, are inherited from the RG equation. • 11B7n(B17BF|!J6b!K(B B6MLZ!!?B!!!J5~ET650iBg3X?t3X2J!K(B Non-connected bifurcating cycles between equi-dimension and hetero-dimension B%"%V%9%H%i%/%H!'(B This is the joint research with H. Kokubu(Kyoto Univ.) and M.-C. Li(NCTU) about dynamics of three-dimensional diffeomorphisms having some codimension-two bifurcations of heteroclinic cycles. We first present an existence theorem for the blender structure near a non-connected heterodimensional cycle. Using this result, we show that diffeomorphisms having the codimension-two bifurcating cycle are contained in the common boundary of the class of diffeomorphisms having equidimensional cycles and of the class of diffeomorphisms that are approximated by non-connected and connected heterodimensional cycles. • 11B7n(B10BF|!J6b!K(B BK>7nFX;K!J4pAC@8J*3X8&5f=j!&M}O@@8J*3X8&5fItLg!K(B B0dEA;R@)8f%M%C%H%o!<%/N9=B$$H:YK&>uBVB?MM@-$K$D$$F(B • 10B7n(B27BF|!J6b!K(B Peter Haissinsky (Universite de Provence) A new characterisation of semi-hyperbolic rational maps Abstract: Semi-hyperbolic rational maps form a particularly interesting class for they may be characterised in many different ways. In this talk, a characterisation using hyperbolic geometry will be proposed, which emphasises their relationship to convex cocompact Kleinian groups. • 10B7n(B20BF|!J6b!K(B B0p@8(B B7<9T!J5~ETBg3XBg3X1!M}3X8&5f2J!K(B Discontinuity of straightening maps for renormalizable polynomials • 10B7n(B13BF|!J6b!K(B B1'I_=EW"!J5~ETBg3XBg3X1!?M4V!&4D6-3X8&5f2J!K(B Siegel disks, Siegel Reinhardt domains and KAM matters in complex Henon maps • 10B7n(B6BF|!J6b!K(B Markus Eiswirth (Fritz-Haber Inst. Berlin) Convex and Toric Geometry for Chemical Reaction Systems Abstract: The steady state and stability problems of complex reaction networks with nonlinear kinetics can be addressed by reformulation in convex coordinates and by using the properties of certain algebraic structures defined on the reaction monomials. The procedure is exemplified using examples of bistable and oscillatory surface reations, showing which features in a network can give rise to instabilities. • 7B7n(B28BF|!J6b!K8a8e(B3B;~hj(B Rich Stankewitz (Ball State University, U.S.A.) Iteration, random dynamics, and polynomial semigroups. Abstract: (Joint work with H. Sumi (Osaka University)) In this talk we will introduce the complex dynamical systems area of study known as the dynamics of rational semigroups. This area can be thought of as an extension of the classical iteration theory. We will also see that it is intimately related to the so-called random iteration theory. In the talk we will introduce and discuss some fundamental properties of the sets of stability (Fatou set) and chaos (Julia set). Our main goal will be to investigate polynomial semigroups with bounded postcritical set in the plane, i.e., G will be a semigroup of complex polynomials (under the operation of composition of functions) such that there exists a bounded set in the plane which contains any finite critical value of any map g \in G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this talk, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G. Furthermore, we provide results concerning the (semi) hyperbolicity of such semigroups as well as discuss various topological consequences of the postcritically boundedness condition. • 7B7n(B14BF|!J6b!K(B Pieter Collins (CUI) Control and Verification of Hybrid-Time Systems Michal Misiurewicz (IUPUI) Double standard maps Abstract: We investigate the family of double standard maps of the circle onto itself, given by f_{a,b}(x)=2x+a+(b/\pi)\sin(2\pi x) (mod 1), where the parameters a,b are real and 0\le b\le 1. Similarly to the well known family of (Arnold) standard maps of the circle, A_{a,b}(x)=x+a+(b/2\pi)\sin(2\pi x) (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a,b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues, that begin at the level b=0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior. This is joint work with A. Rodrigues. • BNW;~NO3X7O%;%_%J!<(B 6B7n(B22BF|!JLZ!K8a8e(B3B;~H>+i(B5B;~H>(B B5~ETBg3XBg3X1!M}3X8&5f2J(B1B9f4[(B5B3,(B 516B?t3XBg9V5A<<(B (B$$$D$b$HF|;~!&>l=j$,0c$^$9(B)
Howard Masur (University of Illinois at Chicago)
Ergodic Theory of translation surfaces

Abstract:
Let X be a closed Riemann surface and omega a holomorphic 1 form on X. The pair (X,(J\(Bomega) defines the structure of a translation surface. This structure is equivalent to one that is given by a collection of polygons in the plane that are glued along their boundaries by translations. For each direction theta, there is a flow in direction theta by straight lines on the surface. In genus one this gives the well known linear flow on the torus. In higher genus there are many additional interesting phenomena. This talk will survey what is known about the ergodic theory of these flows.

• 6$B7n(B16$BF|!J6b!K(B
$B@DLx(B $BIY;o@8!J5~ETBg3XBg3X1!>pJs3X8&5f2J!K(B
$B?@7P%M%C%H%o!<%/$NM}O@%b%G%k!$%@%$%J%_%/%9$N=EMW@-(B • 6$B7n(B2$BF|!J6b!K(B $B?y;3(B $BEP;V!J5~ETBg3XBg3X1!M}3X8&5f2J!K(B moduli space of polynomial maps • 5$B7n(B26$BF|!J6b!K(B $BBgNS(B $B0lJ?!J5~ETBg3XBg3X1!M}3X8&5f2J!K(B 1$B
• 5$B7n(B19$BF|!J6b!K(B
$B0p@8(B $B7<9T!J5~ETBg3XBg3X1!M}3X8&5f2J!K(B
$B$/$j$3$_2DG=$J(B3$B • 4$B7n(B28$BF|!J6b!K(B $B3$BBNLdBj$N?6F02r(B

$BO"Mm@h!'(B $B0p@8(B $B7<9T(B $B")(B606-8502 $B5~ET;T:85~6hKLGr@nDIJ,D.(B $B5~ETBg3XBg3X1!M}3X8&5f2J(B \$B?t3X65<<(B