# Abstracts of Dynamical Systems Research Papers

See also

### Golden gaskets: variations on the Sierpinski sieve

(with Dave Broomhead and Nikita Sidorov)

*Nonlinearity* **14** (2004), 1455-1480.

We consider the iterated function systems (IFSs) that
consist of three general similitudes in the plane with centres
at three non-collinear points, and with a common contraction
factor λ in (0,1).

As is well known, for λ=1/2 the invariant set,
S_{λ}, is a fractal called the Sierpinski
sieve, and for λ<1/2 it is also a fractal. Our goal
is to study S_{λ} for this IFS for
1/2<λ<2/3, i.e., when there are "overlaps" in
S_{λ} as well as "holes". In this
introductory paper we show that despite the overlaps (i.e.,
the Open Set Condition breaking down completely), the
attractor can still be a totally self-similar fractal,
although this happens only for a very special family of
algebraic λ's — the so-called *multinacci numbers*. We
evaluate dim_{H}(S_{λ}) for these
special values by showing that S_{λ} is
essentially the attractor for an infinite IFS which does
satisfy the Open Set Condition. We also show that the set
of points in the attractor with a unique "address" is
self-similar, and compute its dimension.

For "non-multinacci" values of λ we show that if
λ is close to 2/3, then S_{λ} has a
nonempty interior and that if λ<1/sqrt(3) then
S_{λ} has zero Lebesgue measure. Finally we
discuss higher-dimensional analogues of the model in
question.

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(2.8Mb)

### Group theoretic conditions for existence of robust relative
homoclinic trajectories

(with Peter Ashwin)

*Math. Proc. Camb. Phil. Soc.* **133** (2002), 125-141.

We consider robust relative homoclinic trajectories (RHTs) for
*G*-equivariant vector fields. We give some conditions on the
group and representation that imply existence of equivariant vector
fields with such trajectories. Using these result we show very
simply that abelian groups cannot exhibit relative homoclinic
trajectories. Examining a set of group theoretic conditions that
imply existence of RHTs, we construct some new examples of robust
relative homoclinic trajectories. We also classify RHTs of the
dihedral and low order symmetric groups by means of their
symmetries.

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(250K)

### Real continuation from the complex quadratic family: Fixed point
bifurcation sets

(with Bruce Peckham).

*Int. J. Bifurcation
and Chaos* **10** (2000), 391-414

We study the family of real maps of the plane given by *
f(z) = z*^{2} + z + C + Az_bar . For *A=0* this is
the usual complex squaring family (up to coordinate change). As
*A* is varied, the fixed points of the complex squaring map
undergo certain bifurcations. In particular, at *(C,A) =
(0,0)* the map has a very degenerate fixed point at *z=0*,
with a semisimple eigenvalue 1 of multiplicity 2. This is then
unfolded by varying the parameters A and C.

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480K)

### Bifurcation générique d'ondes rotatives d'isotropie
maximale.

(with Pascal Chossat and Muriel Koenig),

*C.R.Acad. Sci. Paris * **320** Série I (1995), 25-30.

We consider bifurcations of relative equilibria (or rotating
waves), using orbit space methods.

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James Montaldi