Positivity 3: 149–172, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
On the Dynamics of Homeomorphisms on the Unit
Ball of R
NILSON C. BERNARDES JR.
Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n,
24020-140, Niterói, RJ, Brasil.
(Received: 16 August 1998; Accepted 16 August 1998)
Abstract. Consider a compact convex subset X of R
(n 2) with non-empty interior and let
H(X)be the set of all homeomorphisms from X onto X endowed with the supremum metric. We are
interested in studying the dynamics of functions in H(X)from the following point of view: Which
properties are satisﬁed by “most” functions in H(X), in the sense that the set of all functions in H(X)
that do not satisfy the given property is of the ﬁrst category? We prove that most functions in H(X)
have uncountably many periodic points of period m, for each m 1, but have no attractive cycles.
Also, for most functions f ∈ H(X), the set of all periodic points of f has no isolated points, is
nowhere dense, has inﬁnitely many connected components, is nowhere closed, is dense in the set of
all non-wandering points of f , and has Lebesgue measure zero. Moreover, most functions in H(X)
are not sensitive to initial conditions on any subset of X that is somewhere dense, but are sensitive
to initial conditions on an uncountable closed connected subset of X. Finally, we prove that most
functions in H(X)have inﬁnitely many pairwise disjoint uniform attractors with certain properties,
but have no attractors with a dense orbit (hence, no strange attractors).
Mathematics Subject Classiﬁcation (1991): 54H20
Key words: Homeomorphisms, orbits, periodic points, sensitivity to initial conditions, attractors
Throughout we ﬁx an integer n 2 and a compact convex subset X of R
non-empty interior. Moreover, we denote by H(X)the set of all homeomorphisms
from X onto X endowed with the supremum metric
d(f,g) := sup
f(x)− g(x) (f, g ∈ H (X)),
where ·denotes the Euclidean norm in R
Since the study of dynamical systems can be thought of as a study of very gen-
eral laws of nature, it should be natural to consider results which are true for “most
transformations”. This can obviously be done in several different contexts. Also,
there are different ways to deﬁne what should be meant by “most transformations”.
In the present work, we restrict ourselves to the study of the dynamics of functions
in H(X). Moreover, the expression “most functions in H(X) have property P ”