And as you know, Jim is extremely American. We worked on many projects. One project was on one-dimensional maps which were expanding. So I was able to do the algebra to make it work and we got our first paper in on expanding maps and chaos. So that was my first chaos paper. Celso Grebogi Institute for Complex Systems and Mathematical Biology, University of Aberdeen, UK — I wrote something that I should say tonight, but now it does not sound it right in view of how this evening is unfolding.

Jim, what is 12 How Did You Get into Chaos so typical of him, had this wonderful idea for us to recount how we did start working on chaos. It is proper for me then to relate too my first encounter with chaos. I went to Berkeley as a post-doc in September to work with Allan Kaufman. I should also mention that I am deeply grateful for Allan and his wife Louise for taking the time and making the effort to come to Aberdeen to be with us.

Back to chaos, when I got in Berkeley in , Allan has just received from Boris Chirikov an advanced copy of his path-breaking paper which was eventually published in Physics Reports in In order to go over the paper, Allan set up round table discussion sessions that met every Thursday, for the whole afternoon. As a curiosity, often a bright and young looking fellow would come to the meetings and would answer all the difficult mathematical aspects of the paper; soon we found out that he was Alan Weinstein.

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Oscar Lanford was another person that often helped us with the mathematics of the theory of dynamical systems, especially the ergodic theory aspects. Those sessions were later enlarged with the inclusion of Jerrold Marsden, and led to the discussion and development of major projects in dynamics, especially the conservative one.

Those were wonderful years in which I learned some of the mathematics of chaotic dynamics; I spent 3 years in Berkeley before moving to Maryland. Ed Ott has already told you the story of when I got there and initiated the collaboration with him and Jim. It remains for me to express to each one of you how deeply gratified and honoured I am for your presence at this Conference.

Devaney Abstract Our goal in this paper is to describe the dynamical behavior of singular perturbations of complex dynamical systems. Singular perturbations occur when a pole is introduced into the dynamics of a polynomial. We then describe some of the many ways Sierpinski curve Julia sets arise in this family. We also give a classification of the dynamics on these sets and describe the intricate structure that occurs around the McMullen domain in the parameter plane for these maps.

By a singular perturbation we mean the following.

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Suppose we have a complex analytic map F0 which, for simplicity, we will assume to be a polynomial of degree d. So the dynamics of F0 are well understood. There are many other equivalent definitions of the Julia set.

## Fully Chaotic Maps and Broken Time Symmetry

The complement of the Julia set is called the Fatou set. However, there really is only one free critical orbit since, when n is even, both of the critical values are mapped to the same point. Thus this family of maps, like the quadratic polynomial family, is a natural one-parameter family of maps.

Consider the second iterate of the critical points. The reason this is significant will become clear in the next section. However, the cycles involved may be different depending on k and, indeed, they may even have different periods. Nonetheless, all points lying on this 16 R. We say that these sets possess 2n-fold symmetry.

See [1] for a proof of this fact. Then: 1. A Sierpinski curve is a very interesting topological space. By definition, a Sierpinski curve is a planar set that is homeomorphic to the well-known Sierpinski a. Devaney carpet fractal. But a Sierpinski curve has an alternative topological characterization: any planar set that is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by disjoint simple closed curves is known to be homeomorphic to the Sierpinski carpet [2].

### Discrete, Nonlinear and Disordered Optics

Moreover, such a set is a universal planar set in the sense that it contains a homeomorphic copy of any compact, connected, one-dimensional subset of the plane. We remark that the second part of the Escape Trichotomy was first proved by McMullen [3]. In Fig. The other disks in the connectedness locus correspond to Sierpinski holes in which the corresponding Julia sets are Sierpinski curves. It follows easily that each of the Ij is mapped univalently onto a region that contains all of the Ik. But then the Riemann — Hurwitz formula implies that A must be an annulus that contains all of the critical points.

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## CiNii Articles -

In this case the Julia set is a Sierpinski curve. To show this, we need to verify the five properties that characterize a Sierpinski curve. It turns out that four and a half of these preperties are trivial to show. This is the only non-standard portion of the proof. However, assuming these boundaries are indeed simple closed curves, they cannot intersect because any such intersection point would necessarily be a critical point. For the proof that these boundaries are simple closed curves, we refer to [1].

## Complex Systems

The following result is proved in [5, 6]. Theorem 2 There is a unique center of each Sierpinski hole.

The proof of this result uses quasiconformal surgery techniques to show that there is a unique center of each Sierpinski hole. All of the parameters from this large collection of holes thus have Julia sets that are homeomorphic, so the natural question is: are the dynamics on these Julia sets the same? The proof of the first part of this theorem follows by quasiconformal surgery techniques. Hence, two such conjugate maps must have the same escape times. Finally, for part three, it suffices to consider the maps that are the centers of the corresponding holes. Similarly, there are , , Sierpinski holes with escape 22 R.

Devaney time 13 but only 60, , different conjugacy classes, so clearly there is a great variety of different dynamical behaviors on these escape time Sierpinski curve Julia sets. The reason for the different number of conjugacy classes when n is even and odd comes from the fact that, when n is odd, there are no Sierpinski holes that meet the real axis and so have no comple conjugate holes.

Along the real axis there is only a pair of Mandelbrot sets and the McMullen domain. See Fig. When n is even, the situation is very different; there is always a Cantor necklace along the negative real axis more about this in Sect. See Figs. The central disk is the McMullen domain M. The entire region in the parameter plane for which this occurs is called the McMullen domain, M. It is known [9] that M is an open, simply connected region that is bounded by a simple closed curve.

In this section we describe some of the remarkable structure that surrounds M in the parameter plane.

Closer inspection seems to indicate that these curves also pass through small copies of Mandelbrot sets as well. This is indeed true, as the following result was shown in [5, 10]. By a center of a baby Mandelbrot set, we mean the parameter drawn from the main cardioid of the Mandelbrot set for which the corresponding attracting cycle is actually superattracting, i. It turns out that all of these baby Mandelbrot sets are buried in the sense that they do not touch the outer boundary of the connectedness locus in the parameter plane.

Then it is known that any parameter drawn from the main cardioid of this Mandelbrot set has a Julia set that is also a Sierpinski curve. For a proof that parameters from the main cardioids of buried baby Mandlebrot sets also yield Sierpinski curves, see [5, 11]. Here are some of the ideas involved in the proof of the rings around M theorem.

This is a map that takes the parameter plane to the dynamical plane. This map takes the sector in the parameter plane to itself. A Cantor necklace is a planar set that is homeomorphic to the following set. Consider the Cantor middle-thirds set lying in the unit interval. Replace each removed open interval with a circular open disk whose diameter is the same as the length of the removed interval and whose boundary touches the two endpoints of the removed interval.

The union of the Cantor set with these countably many open disks is a Cantor necklace. Then consider the following two regions I0 and I1.

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Note the large Sierpinski hole along the negative real axis flanked by two smaller Sierpinski holes which are, in turn, each flanked by two smaller Sierpinski holes, etc 26 R. Devaney portion of the Cantor necklace corresponding to parameters for which the critical orbits land on points in I1. We no longer have a McMullen domain. The large central region is not a McMullen domain; rather it is a Sierpinski hole and it does not contain the origin.

We sketch the proof of this. Also, the second image of all of the critical points is given by Fig. Since the escape times of these Sierpinski holes are all different, it follows that any two parameters drawn from diferent holes in this collection have non-conjugate dynamics as shown in Section 5. For example, in Fig. This first part of this result is somewhat surprising, since it is well known that Julia sets can never contain open sets unless the Julia set is the entire Riemann sphere.

The reason why these Julia sets converge to the closed unit disk D is as follows.