1.2. THE
when s / Fn. Let x U E0 and note that
d (fs(x ),fs(χ(x )))
for all s / Fn because (x , χ(x )) An, cf. (1.7), and that
d (fs(χ (x )),fs(χ(x ))) δ
for all s Fn because χ(x ),χ (x ) W . It follows that
d (fs(χ (x )),fs(χ(x ))) δ
for all s S, and hence that χ(x ) = χ (x ) because of (1.4). Since U E0
is dense in U we conclude that χ(x ) = χ (x ). Thus (x , y ) = (x , χ(x ))
{(z, χ(z)) : z W }, completing the proof of (1.9).
It follows from (1.7) and (1.9) that ξ
) × W
An Rf (X, E)
Ω An Rf (X, E), proving that Ω An Rf (X, E) is open in the topology of
An Rf (X, E) inherited from E × E.
Remark 1.13. As observed in Example 1.1 an expansive homeomorphism ϕ
of a compact metric space (X, d) gives rise to a relatively expansive system in a
canonical way. For such a system it is clear that conjugacy of two points x, y X
implies that x and y are homoclinic in the sense that
= 0.
In many cases, such as the Smale spaces of Ruelle, this condition is sufficient to
ensure the existence of a local conjugacy between x and y; that is, points are
conjugate if and only if they are homoclinic.
To give an example with two homoclinic points that are not locally conjugate,
consider the even shift Y which is the two-sided shift in the alphabet {0, 1} obtained
by disallowing the words
: k = 0, 1, 2,... . Set yi = 0,i Z, and xi =
0,i Z\{0}, x0 = 1. Then x = (xi)i∈Z and y = (yi)i∈Z are both elements of Y , and
x and y are homoclinic under the shift. To see that x and y are not conjugate, let
δ 0 be an expansive constant for Y such that z, z Y, d (z, z ) δ z0 = z0.
Assume to get a contradiction that (U, V, χ) is a conjugacy from x to y. There is
then a K N such that
(1.10) χ(z)k = zk, |k| K, z U.
By definition of the topology of Y there is an open neighborhood V0 of y such that
y V0 V and z[−K,K] = y[−K,K] =
when z V0. Consider the sequence
ai,i N, where
ai = . . .
. . .
Then ai Y for all i and limi→∞ ai = x. In particular, ai χ−1 (V0) for all i large
enough. For such an i, χ (ai)[−K,K] = 02K+1 and χ (ai)k = (ai)k for all |k| K,
thanks to (1.10). In particular, for some i K,
χ (ai) =
which is not an element of Y . It follows that x and y are not conjugate.
Remark 1.14. This remark concerns the relationship between the construction
of Section 1.2 and a construction of Renault, cf. p. 139 of [Re1], which has subse-
quently been developed further by himself as well as by Deaconu, Anantharaman-
Delaroche and others. In the most general setup (with compact unit space) the
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