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A. S. Closed Subgroups of the Solenoid
A.6. Topological Properties of the Solenoid
1. Ultrametric Spaces
1.1. Ultrametric Distances
Let (X, d) be a metric space. Thus X is equipped with a distance function d : X x X --) R>0 satisfying the characteristic properties
d(x,y)>0x#y,
d(y, x) = d(x, y),
d(x, y) < d(x, z) + d(z, y) for all x, y, and z E X. For r > 0 and a E X we define]
B<r(a)_(xEX:d(x,a)<r}
= dressed ball of radius r and center a,
1 Let me use this unconventional terminology in this section only. From (11.2) on, I shall rely on the reader for a proper distinction between "open" and "closed" balls.
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B<r(a) = {x E X : d(x, a) < r}
= stripped ball of radius r and center a.
Hence B<r(a) is empty if r = 0, and the stripped balls form a basis of a topology on X: In particular, all stripped balls are open.
Definition. An ultrametric distance on a space X is a distance (or metric) satisfying the strong inequality
d(x, y) < max(d(x, z), d(z, y)) (< d(x, z) + d(z, y) )
for all x, y, and z E X. An ultrametric space (X, d) is a metric space in which the distance satisfies this strong inequality.
The following results are valid in ultrametric spaces.
Lemma 1. (a) Any point of a ball is a center of the ball.
(b) If two balls have a common point, one is contained in the other.
(c) The diameter of a ball is less than or equal to its radius.
PROOF (a) If b E B<r(a), then d(a, b) < r and
x E B<r(a)
d(x, a) < r
d( rd(x, b) < r x E B<r(b) proving B<r(a) = B<r(b). The case of a dressed ball is similar.(b) Take, for example, a common point c of the balls B<r(a) and B<r'(b). By the previous part, we have
B<r(a) = B<r(c) and B<r'(b) = B<r'(C).
Now, it is clear that B<r(C) C B<r'(c) if r < r', while B<r'(c) C B<r(C) if r' < r.
All other cases are treated similarly. Part (c) is obvious.
It is immediately seen by induction that ultrametric distances also satisfy the strong inequality for finite sequences xl, x2, ... , x E X:
d(xt,x,,) < max(d(xi, x2), d(x2, x3), ..., xn))-
Consider a cycle containing n > 3 distinct points: xi (1 < i < n), xt. We may assume d(xi, maxi <n d(x,, x,+i ): Renumber these points if necessary, and observe that d(x,,, d(x,,, xi) = d(xi, Since
d(xi, xn) < max (d(xi, x2), - .. , xn)) by the ultrametric inequality, it follows that
d(xt, d(xi, xi+i)
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1. Ultrametric Spaces 71
for at least one index 1 < i < n - 1. In other words, the cycle has at least two pairs of consecutive points with equal maximal distance. In particular, in a set a, b, c of cardinality 3, at least two pairs have the same (maximal) length. A picturesque way of formulating this property is this:
In an ultrametric space, all triangles are isosceles (or equilateral), with at most one short side.
Here is an image of the situation. Let x be the earth and y, z be two stars in a galaxy not containing the earth, so that d(x, y) > d(y, z). Then we consider that d(x, y) = d(x, z) (this is the distance of the galaxy containing y, z to the earth).
In other words, ultrametric distances behave as orders of magnitude.
Let us denote by S,(a) = {x E X : d(x, a) = r} the sphere of center a and radius r > 0. Then if a ball B does not contain the point a, it lies on the sphere Sr(a), where r = d(a, B)
if B = B<s(b), then r = d(a, b) > s and B C Sr(a), and similarly,
if B = B<S(b), then r = d(a, b) > s and B C Sr(a).
Let us reformulate these properties in the form of another lemma.
Lemma2. (a) If d(x, z) > d(z, y), then d(x, y) = d(x, z).
(b) If d(x, z) d(z, y), then d(x. y) = max (d(x, z), d(z, y)) (c) If x E Sr(a), then B<r(x) C Sr(a) and
Sr(a) = U B<r(x)
xES,(a)
Balls within a ball
The stripped balls are open in any metric space: By definition, they make up a basis of the topology. Similarly, the dressed balls are closed in any metric space.
In an ultrametric space we have some other peculiarities.
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Lemma 3. (a) The spheres Sr(a) (r > 0) are both open and closed.
(b) The dressed balls of positive radius are open.
(c) The stripped balls are closed.
(d) Let B and B' be two disjoint balls.
Then d(B, B') = d(x, x') for any x E B. X' E B'.
PROOF (a) The spheres are closed in all metric spaces, since the distance function x N d(x, a) is continuous. A sphere of positive radius is open in an ultrametric space by part (c) of the previous lemma.
(b) If r > 0, then B<r(a) = B<r(a) U Sr(a) is open.
(c) If r > 0, the sphere Sr(a) is open; hence B<r(a) = B<r(a) - Sr(a) is closed.
If r = 0, B<r(a) = 0 is closed.
(d) Take four points: x, y E B and x', y' E B'. The 4-cycle of points x, x', y', y has two pairs with maximal distance: They can only be d(x, x') = d(y, y'), since we assume that the balls are disjoint. All pairs of points x E B, x' E B' are at the same distance, and d(B, B') := d(x, x') is this common value.
Due to the frequent appearance of simultaneously open and closed sets in ultra- metric spaces, it is useful to introduce a definition.
Definition. An open and closed set will be called a clopen set.
Lemma 4. (a) A sequence (xn)n,o with d(xn, xn+i) - 0 (n --> oo) is a Cauchy sequence.
(b) If xn -* x a, then d(xn, a) = d(x, a) for all large indices n.
PROOF (a) Observe that if d(xn, xn+i) < E for all n > N, then also d(xn, xn+m) < max d(xn+i, xn+i+1) < E
0<i <m
for all n > N and m > 0.
(b)In fact, d(xn,a)=d(x,a)as soon as d(xn,x)<d(x,a).
Proposition. Let Q C X be a compact subset.
(a) For every a E X - Q, the set of distances d(x, a) (x c S2) is finite.
(b) For every a E S2, the set of distances d(x, a) (x E Q - (a}) is discrete
in R>0.
PROOF (a) We have just seen that
d(x, y) < d(x, a) = d(y, a) = d(x, a);
hence the function f : x H d (x, a), S2 -- R,o is locally constant and continuous.
Its range is finite: The sets f -1(c) (for c E f (S2)) form an open partition of the compact set Q.
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1. Ultrametric Spaces 73
(b) The map f : x H d(x, a), Q - {a} -+ R>o is locally constant as before.
For s > 0, its restriction to the compact subset Q - B«(a) has finite range. This proves that all sets
[E, co) fl {d(x, a) : x E S2, x a}
are finite. Hence f (S2 - {a}) is discrete in c R>0.
Let us summarize.
Properties of ultrametric distances.
(a) Any point of a ball is a possible center of the ball
b E B<r(a) = B<r(b) = B<r(a) (and similarly for stripped balls).
(b) If two balls have a common point, then one is contained in the other.
(c) A sequence (x is a Cauchy sequence precisely when d(x,,, 0 (n -> oo).
(d) In a compact ultrametric space X, for each a E X,
the set of nonzero distances {d(x. a) : a x E X) is discrete in R>0.