Valuations and hyperbolicity in dynamics

55 Pages·2001·0.421 MB·English

by  Ward T.

VALUATIONS AND HYPERBOLICITY IN DYNAMICS
THOMAS WARD
PRODYN Summer School June-July 2001
Georg-August-Universit¨at G¨ottingen
Contents
1. Introduction............................................... 2
2. S-integer dynamical systems............................... 3
2.1. Definition and examples 3
2.2. Background on adeles 7
2.3. Adelic covering space 8
2.4. Topological entropy 10
2.5. Dynamical properties 12
2.6. Periodic points 13
2.7. Growth rates 15
2.8. Typical group automorphisms 19
3. Bernoullicity and recurrence............................... 24
3.1. Automorphisms of solenoids 25
3.2. Exponential recurrence 27
3.3. Commuting automorphisms 27
4. Mixing.................................................... 28
4.1. Background from algebra 30
4.2. Order of mixing – connected case 32
4.3. Order of mixing – disconnected case 33
4.4. Typical actions 40
5. Subdynamics.............................................. 40
5.1. Examples 44
5.2. Adelic amoebas 47
6. Some directions for future research ........................ 49
6.1. Typical group automorphisms 49
6.2. Periodic points 49
6.3. Mixing problem 50
6.4. Entropy 50
6.5. Entropy and Deligne periods 50
References..................................................... 51
1
2 THOMAS WARD
1. Introduction
One of the most basic dynamical ideas is that of a local portrait
of hyperbolicity (or non-hyperbolicity). This is a picture of how the
map acts in a neighbourhood of a point (or, equivalently, on a covering
space).
Example 1.1. [A contracting homothety] Consider the map on
(cid:18) (cid:19) (cid:18) (cid:19)
x λx
R2 given by f : 7→ with λ ∈ (0,1). The local portrait
y λy
Figure 1 around the fixed point 0 shows the dynamics of iterating f:
all orbits are sucked exponentially towards 0.
A (cid:1)
@ A ? (cid:1) (cid:0)
AU (cid:1)(cid:11)
H @@R (cid:0)(cid:9)(cid:0) (cid:8)
H @ A (cid:1) (cid:0) (cid:8)
Hj (cid:8)(cid:25)
H@A (cid:1)(cid:0)(cid:8)
H (cid:8)
- H@A(cid:8)(cid:1)(cid:0) (cid:27)
(cid:8)H
(cid:8)(cid:0)(cid:1)A@H
(cid:8) H
(cid:8)* (cid:0)(cid:1) A@ HY
(cid:8) H
(cid:8) (cid:0) (cid:1) A @ H
(cid:0)(cid:0)(cid:18) @I@
(cid:1)(cid:21) AK
(cid:0) (cid:1) 6 A @
(cid:1) A
Figure 1. A contracting homothety
A more realistic example is given by a hyperbolic toral automor-
phism.
(cid:18) (cid:19) (cid:18) (cid:19)
x 2 1
Example 1.2. Consider the map f : 7→ on R2. Figure 2
y 1 1
shows the eigenvectors in bold, and orbits of points being attracted to
the unstable direction.
Our purpose in these notes is to explore several questions.
(1) In which dynamical settings can these kind of portraits be use-
fully made?
(2) More generally, in which dynamical settings does the action
seen through a valuation tell you anything?
(3) Finally, can valuations in low-dimensional systems help us to
understand actions of higher-rank groups?
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 3
J
J
JJ^ (cid:17)(cid:17)
(cid:17)
J (cid:17)
J (cid:17)(cid:17)3
(cid:17)
JJ(cid:17)
(cid:17)
(cid:17) JJ
(cid:17)+(cid:17) J
(cid:17)
(cid:17) J
(cid:17)(cid:17) J]J
J
J
Figure 2. A hyperbolic automorphism
2. S-integer dynamical systems
2.1. Definition and examples. TheS-integerdynamicalsystemsare
a very simple collection of dynamical systems which are the pieces from
whichgroupautomorphismsmaybebuiltup. Mostofthematerialhere
is taken from [11]. An excellent modern treatment of Tate’s thesis and
related material is the text of Ramakrishnan and Valenza, [57].
Let k be an A–field in the sense of Weil (that is, k is an algebraic
extension of the rational field Q or of F (t) for some rational prime
q
q), and let P(k) denote the set of places of k. A place w ∈ P(k) is
finite if w contains only non–archimedean valuations and is infinite
otherwise (with one exception: for the case F (t) the place given by
p
t−1 is regarded as being an infinite place despite giving rise to a non–
archimedean valuation).
Example 2.1. For the case k = Q or k = F (t), the places are
0 0 q
defined as follows.
The Rationals Q. TheplacesofQareinone–to–onecorrespondence
with the set of rational primes {2,3,5,7,...} together with one addi-
tional place ∞ at infinity. The corresponding valuations are |r| = |r|
∞
(the usual archimedean valuation), and for each p, |r| = p−ordp(r),
p
where ord (r) is the (signed) multiplicity with which the rational prime
p
p divides the the rational r.
The Function Field F (t). For F (t) there are no archimedean
q q
places. For each monic irreducible polynomial v(t) ∈ F [t] there is a
q
distinct place v, with corresponding valuation given by
|f| = q−ordv(f)·deg(v),
v
where ord (f) is the signed multiplicity with which v divides the ra-
v
tional function f. There is one additional place given by v(t) = t−1,
4 THOMAS WARD
and this place will be called an infinite place even though the corre-
sponding valuation is non–archimedean. This ‘infinite’ place is defined
by |f| = q−ordt(f(t−1)).
∞
Let k be a finite extension of k . A place w ∈ P = P(k) is said to lie
0
above a place v of k = Q or F (t), denoted w|v, if |·| rectricted to
0 q w
the base field k ⊂ k coincides with | · | . Denote by k the (metric)
0 v w
completion of k under the metric d (x,y) = |x−y| on k. The local
w w
degree is defined by d = [k : (k ) ]. Choose a normalized valuation
w w 0 v
|·| corresponding to the place w to have
w
|x| = |x|dw/d
w v
for each x ∈ k \{0}, where d = [k : k ] is the global degree. With the
0 0
above normalizations we have the Artin product formula [80, p. 75]
Y
(2.1) |x| = 1
w
w∈P(k)
for all x ∈ k\{0}.
For each finite place w of k, the field k is a local field, and the
w
maximal compact subring of k is
w
r = {x ∈ k: |x| ≤ 1}.
w w
Elements of r are called w–adic integers in k . The group of units in
w w
the ring r is
w
r∗ = {x ∈ k: |x| = 1}.
w w
Let P = P (k) denote the set of infinite places of k.
∞ ∞
Definition 2.2. Letk beanA–field. Givenanelementξ ∈ k∗, andany
set S ⊂ P(k)\P (k) with the property that |ξ| ≤ 1 for all w ∈/ S∪P ,
∞ w ∞
define a dynamical system (X,α) = (X(k,S),α(k,S,ξ)) as follows. The
compact abelian group X is the dual group to the discrete countable
group of S–integers R in k, defined by
S
R = {x ∈ k: |x| ≤ 1 for all w ∈/ S ∪P (k)}.
S w ∞
The continuous group endomorphism α : X → X is dual to the
monomorphism α : R → R defined by α(x) = ξx.
b S S b
Dynamical systems of the form (X(k,S),α(k,S,ξ)) are called S–integer
dynamical systems. Following conventions from number theory, we
shall divide these into two classes: arithmetic systems when k is a
number field, and geometric when k has positive characteristic. To
clarify this definition – and to show how these systems connect with
previously studied ones – several examples follow.
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 5
Example 2.3. (1) Let k = Q, S = ∅, and ξ = 2. Then
R = {x ∈ Q: |x| ≤ 1 for all primes p} = Z,
S p
so X = T and α is the circle doubling map.
(2) Let k = Q, S = {2}, and ξ = 2. Then
R = {x ∈ Q: |x| ≤ 1 for all primes p 6= 2} = Z[1],
S p 2
so X is the solenoid Zd[1], and α is the automorphism of X dual to the
2
automorphism x 7→ 2x of R . This is the natural invertible extension
S
of the circle doubling map [13, Example (c)] or [31, Sect. 2].
As pointed out in [6, Chap. 1 Example D], this dynamical system
is topologically conjugate to the system (Y,β) defined as follows. Let
D = {z ∈ C: |z| ≤ 1} and S1 = {z ∈ C: |z| = 1}. Define a map
f : S1 ×D → S1 ×D by
f(z,ω) = (z2, 1z + 1ω).
2 4
Let Y = T fn(S1 ×D) and let β be the map induced by f on Y.
n∈N
Then there is a homeomorphism Y → X that intertwines the maps β
and α. For more details on this example and related “DE” (derived
from expanding) examples, see, [69, Section I.9]; for a thorough and
detailed treatment of this dyadic example see [34, Sect. 17.1].
(3) Let k = Q, S = {2,3}, ξ = 3. Then R = Z[1], and α is the map
2 S 6
dual to multiplication by 3 on R . This map has dense periodic points
2 S
by [47, Sect. 3] and has topological entropy log3 by [47, Sect. 2].
(4) Let k = Q, S = {2,3,5,7,11,...}, and ξ = 3. Then R = Q and
2 S
α is the automorphism of the full solenoid Qb dual to multiplication by
3 on Q. This map has only one periodic point for any period by [47,
2
Sect. 3] and has topological entropy log3 by [47, Sect. 2].
(5) Let ξ be an algebraic integer, k = Q(ξ) and S = ∅. Then R is
√ p √ S
the ring of algebraic integers in k. Taking ξ = 2 − 1 + i 2 2−2
gives a non–expansive quasihyperbolic automorphism of the 4–torus as
pointed out in [44, Sect. 3]
(6) Let k = F (t), S = ∅, and ξ = t. Then R = F [t], and so
q S q
X = Rc = Q∞ {0,1,...,q − 1}. The map α is therefore the full
S i=0
one–sided shift on q symbols.
(7) Let k = F (t), S = {t}, and ξ = t. Recall that the valuation
q
corresponding to t is |f| = q−ordt(f), so |t| = q−1. The ring of S–
t t
integers is
R = {f ∈ F (t): |f| ≤ 1 for all w 6= t,t−1} = F [t±1].
S q w q
6 THOMAS WARD
The dual of R is then Q∞ {0,1,...,q−1}, and in this case α is the
S −∞
full two–sided shift on q symbols.
(8) Let k = F (t), S = {t}, and ξ = 1 + t. Then X is the two–sided
q
shift space on q symbols, and α is the cellular automaton defined by
(α(x)) = x +x mod q.
k k k+1
(9) Let k = F (t), S = {t,1+t}, and ξ = 1+t. Then α is the invertible
q
extension of the cellular automaton in (8).
(10) Let α be an ergodic automorphism of a finite–dimensional torus.
For each subset S of the rational primes let ΓS = Xb ⊗Z Z[S1]. Then
α defines an endomorphism α : Γc → Γc. Each α has the same
S S S S
entropy as α by [47] (and is therefore measurably isomorphic to α),
but they are all topologically distinct, so {α } forms an uncountable
S
family of topological dynamical systems all measurably isomorphic to
each other.
(11)NotalltoralendomorphismsareS–integerdynamicalsystems. Let
α : Tn → Tn be the toral endomorphism corresponding to the integer
A
matrix A ∈ M (Z). Assume that the characteristic polynomial χ of A
n A
is irreducible, let λ have χ (λ) = 0 and let a = (a ,...,a )t be a vector
A 1 n
in Q(λ)n with Aa = λa with the property that a = a R +···+a R
1 λ n λ
is an ideal in the ring R = Z[λ]. Two ideals determined in this way
λ
from the same matrix belong to the same ideal class by [71, Th. 2].
Lemma 2.4. The toral endomorphism α is topologically conjugate to
the S–integer dynamical system given by k = Q(λ), ξ = λ, S = ∅ if
and only if a defines a trivial element in the ideal class group of R .
λ
Proof. Let B be the companion matrix to the polynomial χ . Then
A
there is an isomorphism from (cid:0)X(k,S),α(k,ξ,S)(cid:1) to (Tn,α ). If a defines
B
a trivial element in the ideal class group of R , then by [71], there is a
λ
matrix S ∈ GL (Z) such that A = SBS−1, so there is an isomorphism
n
from (Tn,α ) to (Tn,α ).
B A
Conversely, let θ : (Tn,α ) → (cid:0)X(k,S),α(k,ξ,S)(cid:1) be a topological con-
A
ˇ
jugacy. Let H denote the first Cech homology functor with coeffi-
1
cients in T; H sends any diagram of solenoids and endomorphisms
1
to an isomorphic diagram by [36, Lemma 6.3]. Then H (θ) defines
1
an isomorphism from (Tn,α ) to (cid:0)X(k,S),α(k,ξ,S)(cid:1); since X(k,S) is an n–
A
dimensional torus, α(k,ξ,S) corresponds to some matrix C ∈ M (Z), and
n
this isomorphism is given by a matrix S ∈ GL (Z) with A = SCS−1.
n
It follows by [71] that a defines a trivial element in the ideal class group
of R . (cid:3)
λ
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 7
2.2. Background on adeles. In this section we assemble some basic
facts about the ring R . For the case S = ∅ most of this is straight-
S
forward. At the opposite extreme, when S contains all finite places (so
R = k), the adelic constructions of [80, Chap. IV] show how to cover
S
the group X(k,S). In the intermediate case, straightforward modifica-
tions of Weil’s arguments are needed. The construction is also given
in Tate’s thesis, and we indicate below how to read off the results we
shall need from this.
Fix an A–field k and a set S of finite places of k.
Definition 2.5. The S–adele ring of k is the ring
( )
Y
k (S) = x = (x ) ∈ k : |x | ≤ 1 for all but finitely many ν ,
A ν ν ν ν
ν∈S∪P∞
with the topology induced by the following property. For each finite
set S0 ⊂ S, the locally compact subring kS0 ⊂ k (S) defined by
A A
Y Y
kS0 = k × r
A ν ν
ν∈S0∪P∞ ν∈S\S0
(with the product topology) is an open subring of k (S), and a fun-
A
damental system of open neighbourhoods of 0 in the additive group of
k (S) is given by a fundamental system of neighbourhoods of 0 in any
A
one of the subrings kS0.
A
Notice that k (S) is locally compact since each r is compact.
A ν
Define a map ∆ : R → k (S) by ∆(x) = (x,x,x,...). This map is
S A
a well–defined ring homomorphism: notice that for α ∈ R , |α| ≤ 1
S ν
for all but finitely many ν by [80, Th. III.1.3].
In [70], Tate introduces the notion of an abstract restricted direct
product, under the hypothesis that P (= S∪P ) is an arbitrary count-
∞
able set of indices (places). Let G (= k ) be a locally compact abelian
P ν
group for P ∈ P, and for all but finitely many P, let H (= r ) be an
P ν
open compact subgroup of G . The restricted direct product is defined
P
as
( )
Y
G(P) = g = (g ) ∈ G : g ∈ H for all but finitely many P ,
P P P P
P∈P
a locally compact abelian topological group. We topologise G(P) by
choosing a fundamental system of neighbourhoods of 1 in G(P) of the
Q
form N = N , where each N is a neighbourhood of 1 in G and
P∈P P P P
N = H for all but finitely many P, which accords with the topology
P P
in Definition 2.5.
8 THOMAS WARD
The key results proved in [70, Lem. 3.2.2, Th. 3.2.1] are the follow-
ing.
(1) ∆(R ) is discrete in k (S) and k (S)/∆(R ) is compact,
S A A S
(2) RS⊥ ∼= RS, k\A(S) ∼= kA(S) and so kA(S)/∆(RS) ∼= RˆS
where S is an arbitrary set of finite places of an A–field k. We collect
these remarks in the following Theorem, which is an extension of one
of the “Main Theorems” in [80, Chap. IV, Sect. 2] to arbitrary sets of
places.
Theorem 2.6. The map ∆ : R → k (S) embeds R as a discrete
S A S
cocompact subring in the S–adele ring of k. There is an isomorphism
between the S–adele ring k (S) and itself, which induces an isomor-
A
phism between RcS and kA(S)/∆(RS).
Remark 2.7. The S–adele ring k (S) covering the dynamical system
A
(X(k,S),α(k,S,ξ)) gives a complete local portrait of the hyperbolicity. A
neighbourhood of the identity in X(k,S) is isometric to a neighbourhood
of the identity in k (S). The map α(k,S,ξ) under this isometry acts
A
on each quasi–factor k by multiplication, dilating the metric on that
ν
quasi–factor by |ξ| . If S is infinite, then the local action is an isometry
ν
on all but finitely many quasi–factors, making such systems very far
from hyperbolic ones.
2.3. Adelic covering space. We first recall how covering spaces arise
for familiar maps. If f : T → T is the doubling map x 7→ 2x mod 1
on the additive circle, then the cover π : R → T lifts the map to
f˜: R → R. Figure 3 shows the lifted map: notice that the projection
π is a local isometry. The import of Section 2.2 is that the same thing
R α˜ - R
π π
? ?
T α - T
Figure 3. Lifting the circle doubling map
happens for any S-integer dynamical system.
Example 2.8. LetαbetheS–integerdynamicalsystemcorresponding
to k = F (t), S = {t} and ξ = t (so the corresponding dynamical
p
system is the full p-shift). The covering space is the product k ×
∞
k where v is the valuation corresponding to t and ∞ the valuation
v
VALUATIONS AND HYPERBOLICITY IN DYNAMICS 9
corresponding to t−1. The local hyperbolicity portrait in the covering
space is shown in Figure 4.
×|t| = p
t−1
6
×|t| = p−1
- (cid:27) t
?
Figure 4. Multiplication by t is hyperbolic for S = {t}
The system is hyperbolic, which shows up in having extremely reg-
ular properties (for example, the dynamical zeta function is rational).
Example 2.9. A non-hyperbolic additive cellular automaton is given
by choosing k = F (t), S = {t} and ξ = 1 + t. This is the additive
p
cellular automata with local rule given by
f(x ,x ) = x +x .
0 1 0 1
If p = 2 this is ‘rule 102’ in the standard description of cellular au-
tomata with radius 1. The covering space is the same product. The
local hyperbolicity portrait is shown in Figure 5, which indicates why
this system is non–hyperbolic.
×|1+t| = p
t−1
6
×|1+t| = 1
t
?
Figure 5. Local effect of multiplication by 1+t
The non-hyperbolicity makes the dynamics extremely complicated:
the direction in which the map acts likes an isometry behaves like a
sort of rotation, destroying some (but not all) periodic points.
The final example is a connected group automorphism.
Example 2.10. Let k = Q, S = {2,3} and ξ = 2. This system is
an isometric extension of the invertible extension of the circle doubling
map. The covering space is R×Q ×Q .
2 3
10 THOMAS WARD
×|2| (cid:17)
∞ (cid:17)×|2|
6 (cid:17) 2
(cid:17)+
(cid:17)
(cid:17)(cid:17) ×|2|3 = 1
(cid:17)(cid:17)
(cid:17)
(cid:17)3
(cid:17)
?
(cid:17)
(cid:17)
Figure 6. Local effect of multiplying by 2 on Zd[1]
6
2.4. Topological entropy. For any automorphism α : X → X of
a compact metrizable group X, the topological entropy h(α) may be
defined in several different ways. The most convenient formulation is
that of Bowen [5], where the topological entropy is expressed as a local
rate of volume growth.
Definition 2.11. The topological entropy of the compact group auto-
morphism α : X → X is defined to be
!
n−1
1 \ (cid:0) (cid:1)
h (α) = limlimsup− logµ α−k B (αkx) ,
Bowen (cid:15)
(cid:15)&0 n→∞ n
k=0
where x is any point, µ is Haar measure, and B denotes the metric
(cid:15)
open ball around x.
Bowen [5, Prop. 7] is that
h(α) = h (α).
Bowen
This gives a very straightforward way to compute the entropy of auto-
morphismsofsolenoids(compact,connected,finite-dimensionalgroups)
– this entropy was computed originally by Yuzvinskii [83], and then a
much simpler proof using Bowen’s formulation and the adelic covering
space was given in [47] and [73] for the solenoid case. The geometric
case, which includes certain cellular automata is similar (see [79]).
Theorem 2.12. The topological entropy of an S–integer system is
given by
X
(2.2) h(α(k,S,ξ)) = log+|ξ|
w
w∈S∪P∞(k)
Proof. The proof is sketched for a simple case. Assume that the field k
has positive characteristic (so all the places are non-Archimedean) and