1.2. THE

´

ETALE EQUIVALENCE RELATION OF LOCAL CONJUGACY 7

when s / ∈ Fn. Let x ∈ U ∩ E0 and note that

d (fs(x ),fs(χ(x )))

δ

2

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 ∈

(

χ−1(W

) × 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

lim

|k|→∞

d

(

ϕk(x),ϕk(y)

)

= 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

102k+11

: 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] =

02K+1

when z ∈ V0. Consider the sequence

ai,i ∈ N, where

ai = . . .

1111102i102i111111

. . .

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) =

1∞04i+11∞

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