357 research outputs found
Checkerboard Julia Sets for Rational Maps
In this paper, we consider the family of rational maps \F(z) = z^n +
\frac{\la}{z^d}, where , , and\la \in \bbC. We consider
the case where \la lies in the main cardioid of one of the principal
Mandelbrot sets in these families. We show that the Julia sets of these maps
are always homeomorphic. However, two such maps \F and are conjugate
on these Julia sets only if the parameters at the centers of the given
cardioids satisfy \mu = \nu^{j(d+1)}\la or \mu = \nu^{j(d+1)}\bar{\la}
where j \in \bbZ and is an root of unity. We define a
dynamical invariant, which we call the minimal rotation number. It determines
which of these maps are are conjugate on their Julia sets, and we obtain an
exact count of the number of distinct conjugacy classes of maps drawn from
these main cardioids.Comment: 25 pages, 14 figures; Changes since March 19 version: added nine
figures, fixed one proof, added a section on a group actio
Dynamical invariants and parameter space structures for rational maps
For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z^n + λ/z^d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z^n. This work presents several results related to these goals for the family Fλ, particularly regarding a structure of "necklaces" in the λ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of "subnecklaces," wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces
Rational maps: the structure of Julia sets from accessible Mandelbrot sets
For the family of complex rational maps F_λ(z)=z^n+λ/z^d, where λ is a complex parameter and n, d ≥ 2 are integers, many small copies of the well-known Mandelbrot set are visible in the parameter plane. An infinite number of these are located around the boundary of the connectedness locus and are accessible by parameter rays from the Cantor set locus. Maps taken from main cardioid of these accessbile Mandelbrot sets have attracting periodic cycles. A method for constructing models of the Julia sets corresponding to such maps is described. These models are then used to explore the existence of dynamical conjugacies between maps taken from distinct accessible Mandelbrot sets in the upper halfplane
Invariant Peano curves of expanding Thurston maps
We consider Thurston maps, i.e., branched covering maps
that are postcritically finite. In addition, we assume that is expanding in
a suitable sense. It is shown that each sufficiently high iterate of
is semi-conjugate to , where is equal to the
degree of . More precisely, for such an we construct a Peano curve
(onto), such that
(for all ).Comment: 63 pages, 12 figure
Trends and Developments in Complex Dynamics
[no abstract available
Combining Rational maps and Kleinian groups via orbit equivalence
We develop a new orbit equivalence framework for holomorphically mating the
dynamics of complex polynomials with that of Kleinian surface groups. We show
that the only torsion-free Fuchsian groups that can be thus mated are punctured
sphere groups. We describe a new class of maps that are topologically orbit
equivalent to Fuchsian punctured sphere groups. We call these higher
Bowen-Series maps. The existence of this class ensures that the Teichm\"uller
space of matings is disconnected. Further, they also show that, unlike in
higher dimensions, topological orbit equivalence rigidity fails for Fuchsian
groups acting on the circle. We also classify the collection of Kleinian Bers'
boundary groups that are mateable in our framework.Comment: 80 pages, 9 figure
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