Carsten Lunde Petersen

Local connectivity of some Julia sets containing a circle with an irrational rotation

The Fatou set FR for a rational map R: C--*C is the set of points z c C possessing a neighbourhood on which the family of iterates {R n }n~>o is normal (in the sense of Montel). The Julia set JR=C--FR is the complement of the Fatou set. (The monographs [CG], [Be], [St] provide introductions to the theory of iteration of rational maps.) Let 0E ]0, 1 [ Q be an irrational number and write it as a continued fraction

Dynamics on the Riemann Sphere

Branner-Hubbard motion is a systematic way of deforming an attracting holomorphic dynamical system f into a family (fs)s∈L, via a holomorphic motion which is also a group action. We establish the analytic dependence of fs on s (a result first stated by Lyubich) and the injectivity of fs on f . We prove that the stabilizer of f (in terms of s) is either the full group L (rigidity), or a discrete subgroup (injectivity). The first case means that fs is Mobius conjugate to f for all s∈L, and it happens for instance at the center of a hyperbolic component. In the second case the map s → fs is locally injective. We show that BH-motion induces a periodic holomorphic motion on the parameter space of cubic polynomials, and that the corresponding quotient motion has a natural extension to its isolated singularity. We give another application in the setting of Lavaurs enriched dynamical systems within a parabolic basin.We consider the Arnold family of analytic diffeomorphisms of the circle x 7! x + t + a 2� sin(2�x) mod (1), where a,t 2 (0,1) and its complexification f�,a(z) = �ze a 2 (z 1 z ) , with � = e 2�it a holomorphic self map of C � . The parameter space contains the well known Arnold tongues Tfor� 2 (0,1) being the rotation number. We are interested in the parameters that belong to the irrational tongues and in particular in those for which the map has a Herman ring. Our goal in this paper is twofold. First we are interested in studying how the modulus of this Herman ring varies in terms of the parametera, when a tends to 0 along the curve T�. We survey the different results that describe this variation including the complexification of part of the Arnold tongues (called Arnold disks) which leads to the best estimate. To work with this complex parameter values we use the concept of the twist coordinate, a measure of how far from symmetric the Herman rings are. Our second goal is to investigate the slice of parameter space that contains all maps in the family with twist coordinate equal to one half, proving for example that this is a plane in C 2 . We show a computer picture of this slice of parameter space and we also present some numerical algorithms that allow us to compute new drawings of non-symmetric Herman rings of various moduli.An exposition of the 1918 paper of Lattès, together with its historical antecedents, and its modern formulations and applications. 1. The Lattès paper. 2. Finite Quotients of Affine Maps 3. A Cyclic Group Action on C/Λ . 4. Flat Orbifold Metrics 5. Classification 6. Lattès Maps before Lattès 7. More Recent Developments 8. Examples References §1. The Lattès paper. In 1918, some months before his death of typhoid fever, Samuel Lattès published a brief paper describing an extremely interesting class of rational maps. Similar examples had been described by Schröder almost fifty years earlier (see §6), but Lattès’ name has become firmly attached to these maps, which play a basic role as exceptional examples in the holomorphic dynamics literature. His starting point was the “Poincaré function” θ : C → Ĉ associated with a repelling fixed point z0 = f(z0 ) of a rational function f : Ĉ → Ĉ . This can be described as the inverse of the Kœnigs linearization around z0 , extended to a globally defined meromorphic function.1 Assuming for convenience that z0 ̸= ∞ , it is characterized by the identity f(θ(t)) = θ(μ t) for all complex numbers t , with θ(0) = z0 , normalized by the condition that θ′(0) = 1 . Here μ = f ′(z0 ) is the multiplier at z0 , with |μ| > 1 . This Poincaré function can be computed explicitly by the formula θ(t) = lim n→∞ f ◦n ( z0 + t/μ n ) . Its image θ(C) ⊂ Ĉ is equal to the Riemann sphere Ĉ with at most two points removed. In practice, we will always assume that f has degree at least two. The complement Ĉ ! θ(C) is then precisely equal to the exceptional set Ef , consisting of all points with finite grand orbit under f . In general this Poincaré function θ has very complicated behavior. In particular, the Poincaré functions associated with different fixed points or periodic points are usually quite incompatible. However, Lattès pointed out that in special cases θ will be periodic or doubly periodic, and will give rise to a simultaneous linearization for all of the periodic points of f . (For a more precise statement, see the proof of 3.9 below.) 1 Compare [La], [P], [K]. For general background material, see for example [M3] or [BM].

No elliptic limits for quadratic maps

We establish bounds for the multipliers of those periodic orbits of

Analytic Coordinates Recording Cubic Dynamics

R_\mu(z) = z(z+\mu)/(1+\overline\mu z)

Dynamics on the Riemann Sphere: A Bodil Branner Festschrift

, which have a Poincare rotation number

Branner–Hubbard motions and attracting dynamics

p/q

On the size of linearization domains

. The bounds are given in terms of

On holomorphic critical quasi circle maps

p/q

On parabolic external maps

and the (logarithmic) hororadius of

On the Julia set of a typical quadratic polynomial with a Siegel disk

\mu