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define Hv = {x " Rd : x · v d" 0} to be the half-space with outward
unit normal v. Let Hd be the set of half-spaces in Rd, which we identify
with Sd-1 via the parameterization v ”! Hv. For H " Hd we denote
its outward unit normal vector by vH.
Expansiveness along a half-space H is defined using Definition 5.3
with F = H. Observe that thickening Hv by t > 0 results merely
in the translation Hv + tv of Hv. Hence there is no need to thicken
half-spaces in the definition, and a Zd-action ² is therefore expansive
along H if and only if there is an µ > 0 such that ÁH(x, y) d" µ implies
²
that x = y.
Definition 5.5. For a Zd-action ² define
E(²) = {H " Hd : H is expansive for ²},
N(²) = {H " Hd : H is nonexpansive for ²}.
An expansive component of half-spaces for ² is a connected component
of E(²).
Remark 5.6. A coding argument analogous to [8, Lemma 3.4] shows
that E(²) is an open set and so N(²) is a compact set.
The following lemma shows that a (d - 1)-plane is nonexpansive
for ² if and only if at least one of the two bounding half-spaces is also
nonexpansive for ². Thus if we define À : Hd ’! Gd-1 by À(H) = "H,
then À(N(²)) = Nd-1(²). This shows that the half-space behavior N(²)
determines the expansive subdynamics of ².
The following key definition is taken from [8, Definition 3.1].
Definition 5.7. Let ² be an expansive Zd-action with expansive con-
stant ´. For subsets E, F of Rd we say that E codes F provided that,
for every x " Rd, if ÁE+x(x, y) d" ´ then ÁF +x(x, y) d" ´.
² ²
Lemma 5.8. Let ² be a Zd-action and V " Gd-1. Then V " Nd-1(²)
if and only if there is an H " N(²) with "H = V .
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 43
Proof. If H " N(²), then V = "H ‚" H is also nonexpansive.
Conversely, let V " Gd-1 and H = Hv, H = H-v be the two half-
spaces with boundary V . Suppose that both H and H are expansive
for ². We prove that V is also expansive for ², which will complete the
proof.
Since ² has an expansive half-space, it is an expansive action. Let
´ > 0 be an expansive constant for ². Let B(r) denote the ball of
radius r in Rd, and [0, v] be the line segment joining 0 to v. A  finite
version of the expansiveness of H, entirely analogous to [8, Lemma 3.2],
is that there is an r > 0 such that H )"B(r) codes [0, v]. Similarly, there
is an s > 0 such that H )" B(s) codes [0, -v]. Hence if t = max{r, s},
t t+1 t+2
then V codes V , which by the same argument codes V , and so
t
on. Thus V codes Rd, which means that V is expansive.
As a starting point, Schmidt [64] gave the following characterization
of expansiveness for ±M. For an ideal p ‚" Rd, let
V(p) = z = (z1, . . . , zd) " (C×)d : f(z1, . . . , zd) = 0 for all f " p .
Let Sd = {(z1, . . . , zd) " Cd : |z1| = · · · = |zd| = 1} be the multiplica-
tive d-torus.
Theorem 5.9. The Zd-action ±M is expansive if and only if both
(1) M is a Noetherian Rd-module, and
(2) for each prime ideal p ‚" Rd associated to M, V (p) )" Sd = ".
The first condition  algebraic in nature  is necessary for the follow-
ing reason. If M is not Noetherian, then there is an infinite ascending
chain of submodules {0} ‚" M1 ‚" M2 ‚" . . . inside M; their annihila-
tors form an infinite descending chain of closed ±M-invariant subgroups
¥" ¥" ¥"
{0}¥" = XM ƒ" M1 ƒ" M2 ƒ" . . . with Mj = {0}, showing that
je"1
±M is not expansive. The second condition  which is geometric  is
necessary because from a point in V (p))"Sd a point may be constructed
whose orbit under the action of ±M stays close to 0.
The main result in [20] is a directional version of this theorem. There
are several steps involved in this, and the two different requirements
for expansiveness each have their own analogues. For H " Hd, define
the ring RH = Z[un : n " H )" Zd], which is a subring of Rd. In general
RH is not Noetherian; indeed, RH is Noetherian exactly when vH is
a rational direction in the sense that RvH )" Zd = {0}, so that RH is
Noetherian for only countably many H.
Theorem 5.10. Let M be a Noetherian Rd-module, ±M be the cor-
responding algebraic Zd-action, and H " Hd. Then the following are
equivalent.
44 THOMAS WARD
(1) ±M is expansive along H.
d
(2) ±R /p is expansive along H for every prime ideal p associated
to M.
(3) Rd/p is RH-Noetherian and [0, ")vH )" log |V(p)| = " for every
p " asc(M).
In order to work with this result, it is important to give a more
computable version of the RH-Noetherian property. This is discussed
in detail in [20] and [21]. From [20] we take the following theorem.
Theorem 5.11. Let M be a Noetherian Rd-module, H " Hd, and
k " Zd H. Then M is RH-Noetherian if and only if there is a
polynomial of the form uk - f(u) with f(u) " RH that annihilates M.
It follows that there is an algorithm that describes the set of those H
for which a given module M is RH-Noetherian.
The last part of this theorem relates to a slightly different kind of
problem than those we have mentioned. That is, given a presentation
of a module, how does one set about actually computing some of the
dynamical properties of the associated system? In particular, for which
properties are complex syzygy computations required? See [22] and [20,
Sect. 6] for some discussion of this.
5.1. Examples. Using the correspondence Hd ”! Sd-1 given by H ”!
vH, subsets of Hd may be identified with the corresponding subsets of
Sd-1. Using this convention, for an ideal a " Rd define
d
Nn(±R /a) = {v " Sd-1 : Rd/a is not RH -Noetherian},
v
d
Nv(±R /a) = {v " Sd-1 : [0, ")v )" log |V(a)| = "}.
d
Observe that Nv(±R /a) is the radial projection of log |V(a)| to Sd-1.
By Theorem 5.10,
d d d
N(±R /a) = Nn(±R /a) *" Nv(±R /a).
In the case of a principal ideal f in Rd we abbreviate V( f ) to V(f).
Example 5.12. Consider Example 4.6 again. As we saw, this has
a surprisingly mixing property, despite having zero entropy. Here
M = R2/ u1 - 2, u2 - 3 ; the corresponding dynamical system ± is
the invertible extension of the semi-group action generated by x ’! 2x
and x ’! 3x mod 1 on the additive circle. Write p = u1 - 2, u2 -
3 . To use Theorem 5.10, notice that V (p) = {(2, 3)}, so the vari-
ety condition [0, ")vH )" log |V(p)| = " will fail only in the direction
vH = (log 2, log 3). The module M is RH-Noetherian except when
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 45
vH = (0, -1) or (-1, 0). A sample of these arguments is the following:
if vH = (-1, 0), then RH is the ring RH = Z[u1, u±1], and so
2
1 1
R2/p ƒ" · · · ƒ" RH/p ƒ" RH/p ƒ" RH/p ƒ"
4 2
is an infinite ascending chain of RH-submodules, showing this direction
to be non-Noetherian. The non-expansive set is shown in Figure 15,
using the convention above associating subsets of the Grassmanian to
subsets of the (d - 1)-sphere. [ Pobierz całość w formacie PDF ]