Slices of parameter space for meromorphic maps with two asymptotic values

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math.101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

FRACTALS FROM NONLINEAR IFSs OF THE COMPLEX MAPPING FAMILY f(z) = zn + c

To generate exotic fractals, we investigate the construction of nonlinear iterated function system (IFS) using the complex mapping family [Formula: see text] ([Formula: see text]). A set of [Formula: see text]-values is chosen from the period-1 bulb of the Mandelbrot set, so that each mapping has an attracting fixed point in the dynamic plane. Computer experiments show that a set of arbitrarily chosen [Formula: see text]-values may not be able to generate a fractal. We prove a sufficient condition that if the [Formula: see text]-values are chosen from a specific region related to a circle in the period-1 bulb, the nonlinear IFS with such complex mappings is able to generate exotic fractal. Furthermore, if the set of [Formula: see text]-values possesses a specific symmetry in the Mandelbrot set, then the fractal also exhibits the same symmetry. We present a method of generating aesthetic fractals with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text] and with [Formula: see text] or [Formula: see text] symmetry for [Formula: see text].

A combinatorial classification of postcritically fixed Newton maps

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to$\infty$through a finite chain of such components.

Bifurcation and Chaos in Synchronous Manifold of a Forest Model

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant [Formula: see text] is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map [Formula: see text] containing two parameters [Formula: see text] and [Formula: see text]. Here [Formula: see text] is the energy depletion quantity and [Formula: see text] is the coupling strength. In particular, we obtain the following results. First, we prove that [Formula: see text] has a chaotic dynamic in the sense of Devaney on an invariant set whenever [Formula: see text], which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that [Formula: see text] exhibits the period adding bifurcation. Specifically, we show that for any [Formula: see text], [Formula: see text] has a unique global attracting fixed point whenever [Formula: see text] ([Formula: see text]) and that for any [Formula: see text], [Formula: see text] has a unique attracting period [Formula: see text] point whenever [Formula: see text] is less than and near any positive integer [Formula: see text]. Furthermore, the corresponding period [Formula: see text] point instantly becomes unstable as [Formula: see text] moves pass the integer [Formula: see text]. Finally, we demonstrate numerically that there are chaotic dynamics whenever [Formula: see text] is in between and away from two consecutive positive integers. We also observe the route to chaos as [Formula: see text] increases from one positive integer to the next through finite period doubling.

Real Dynamics of Family of Functions λxex/(ex-1) for Positive λ

In this paper, the real dynamics of one parameter family of functions hλ(x)=λxex(ex-1), λ>0 is investigated. It is shown that the real fixed point of hλ(x) exists only for 0<λ<1 which is attracting and there is no real fixed point for λ>1 It is found that the whole real line converges to real attracting fixed point of hλ(x) for 0<λ<1.

THE EFFECT OF WEAK DISSIPATION IN TWO-DIMENSIONAL MAPPING

The influence of weak dissipation and its consequences in a two-dimensional mapping are studied. The mapping is parametrized by an exponent γ in one of the dynamical variables and by a parameter δ which denotes the amount of the dissipation. It is shown that for different values of γ the structure of the phase space of the nondissipative model is replaced by a large number of attractors. The approach to the attracting fixed point is characterized both analytically and numerically. The attracting fixed point exhibits a very complicated basin of attraction.

Hausdorff dimension of hairs and ends for entire maps of finite order

We study transcendental entire mapsfof finite order, such that all the singularities off−1are contained in a compact subset of the immediate basinBof an attracting fixed point off. Then the Julia set offconsists of disjoint curves tending to infinity (hairs), attached to the unique point accessible fromB(endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class(i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

Invariant probability measures and dynamics of exponential linear type maps

We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λ<e, there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e-cutoff phenomenon. For deterministic distributions and λ>e, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.

ATTRACTING BASINS OF VOLUME PRESERVING AUTOMORPHISMS OF ℂk

We study topological properties of attracting sets for automorphisms of ℂk. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.