Invariant measures for higher rank hyperbolic Abelian actions

29 Pages·2005·0.65 MB·English

Invariant Measures for Higher Rank Hyperbolic Abelian
Actions
(cid:0) y
A(cid:0) Katok R(cid:0) J(cid:0) Spatzier
Department of Mathematics Department of Mathematics
The Pennsylvania State University University of Michigan
University Park(cid:1) PA (cid:2)(cid:3)(cid:4)(cid:5)(cid:6) Ann Arbor(cid:1) MI (cid:7)(cid:4)(cid:2)(cid:5)(cid:8)
Abstract
WeinvestigateinvariantergodicmeasuresforcertainpartiallyhyperbolicandAnosov
k k k
actions of R (cid:0) Z and Z(cid:0)(cid:1) We show that they are either Haar measures or that every
element of the action has zero metric entropy(cid:1)
(cid:0) Introduction
Actionsofhigherrankabelian groupsandsemigroupsoncompactsmoothmanifoldsdisplay
a remarkable and not yet completely understood array of rigidity properties provided the
action is su(cid:0)ciently hyperbolic(cid:1) Early indications of such phenomena can be found in the
worksof N(cid:1) Koppel and R(cid:1) Sacksteder on commuting one(cid:2)dimensional and expanding maps
(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:5)(cid:8)(cid:1) A(cid:1) Katok and J(cid:1) Lewis established local and global di(cid:9)erential rigidity of the
n(cid:1)(cid:0) n
actions of Z on T by hyperbolic toral automorphisms (cid:3)(cid:4)(cid:4)(cid:8)(cid:1) Some of the phenomena
including trivialization of the (cid:10)rst cohomology group(cid:6) absence of non(cid:2)trivial time changes(cid:6)
local H(cid:11)older and di(cid:9)erential rigidity for a general class of standard abelian actions are
studied in our papers (cid:3)(cid:4)(cid:12)(cid:6) (cid:4)(cid:13)(cid:6) (cid:4)(cid:14)(cid:8)(cid:1) For related developments see (cid:3)(cid:4)(cid:15)(cid:6) (cid:4)(cid:7)(cid:8)(cid:1)
Anotherofthose rigidity properties is therelative scarcity ofinvariant Borel probability
measures(cid:1) It was (cid:10)rst noticed by H(cid:1) Furstenberg in his landmark paper (cid:3)(cid:14)(cid:8) where he posed
the following problem(cid:1)
Furstenberg(cid:0)s Conjecture(cid:1) The only ergodic invariant measures for the semigroup of
n m
circle endomorphisms generated by multiplications by p and q where p (cid:0)(cid:16) q unless n (cid:16)
m (cid:16) (cid:15) are Lebesgue measure and atomic measures concentrated on periodic orbits(cid:0)
Furstenbergestablishes theweakertopological versionofthisstatementbyshowingthat
all topologically transitive sets are either (cid:10)nite or the whole circle(cid:1) This was generalized
(cid:0)
Partially supported by NSF grant DMS (cid:0)(cid:1)(cid:2)(cid:3)(cid:0)(cid:0)(cid:4)
y
Partially supported by the NSF(cid:5)AMS Centennial Fellow
(cid:4)
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:7)
to the optimal results for semigroups of toral endomorphisms by D(cid:1) Berend (cid:3)(cid:7)(cid:6) (cid:12)(cid:8)(cid:1) E(cid:1)
A(cid:1) Satayev proved that the only ergodic invariant measures are Lebesgue and atomic for
largersemigroupsgeneratedbymultiplications byP(cid:17)n(cid:18)n(cid:2)Z(cid:0) whereP isanypolynomialwith
integer coe(cid:0)cients (cid:3)(cid:7)(cid:19)(cid:8)(cid:1) The (cid:10)rst result directly pertaining to Furstenberg(cid:20)s conjecture was
obtained by R(cid:1) Lyons using harmonic analysis(cid:1) He proved that the only invariant measure
which makesthe multiplications exactendomorphisms is Lebesgue (cid:3)(cid:7)(cid:4)(cid:8)(cid:1) D(cid:1)Rudolph and A(cid:1)
Johnson strengthened this result byreplacing the exactness condition with positive entropy
for some and hence all elements of the action (cid:3)(cid:7)(cid:21)(cid:6) (cid:19)(cid:8)(cid:1) At the heart of their arguments lies
(cid:1) (cid:1)
a symbolic version of the natural extension of a Z(cid:2)(cid:2)action to a Z(cid:2)action(cid:1) For further
developments in this speci(cid:10)c problem see (cid:3)(cid:13)(cid:6) (cid:21)(cid:8)(cid:1)
Moregenerally(cid:6) onenoticesasharpcontrastbetweenAnosovdi(cid:9)eomorphismsand(cid:22)ows(cid:6)
i(cid:1)e(cid:1) hyperbolic actions of Zand R(cid:6) which possess an abundance of invariant measures
with very di(cid:9)erent ergodic properties(cid:6) including many measures with positive entropy(cid:6) and
(cid:23)genuine(cid:24) hyperbolic actions of higher rank abelian groups and semigroups(cid:1) In the latter
case(cid:6) all knownergodic invariant measuresare ofalgebraic natureunless(cid:6) like in an example
constructedbyM(cid:1)Reesinanunpublished manuscript(cid:3)(cid:7)(cid:13)(cid:8)(cid:6)thereisaninvariantsubmanifold
on which the action has a factor where it reduces to an action of a rank one group(cid:1) The
question of deciding what hyperbolic (cid:17)Anosov(cid:18) or partially hyperbolic actions should be
considered (cid:23)genuine(cid:24) is rather subtle(cid:1) Obviously(cid:6) in addition to faithfulness one should
require theabsence ofrank onefactorsforthe action and all of its(cid:10)nite covers(cid:1) The central
open question in the area is whether all such actions are of algebraic nature (cid:17)cf(cid:1) (cid:3)(cid:25)(cid:8)(cid:18)(cid:1) For
the time being(cid:6) it is reasonable to list all known examples and bundle them together under
the name of standard actions(cid:1) These include irreducible semigroups of partially hyperbolic
endomorphisms of tori and (cid:17)infra(cid:18)nilmanifolds(cid:6) their natural extensions and suspensions(cid:6)
Weyl chamber(cid:22)owsandrelatedsymmetric spaceexamples andtwistedWeyl chamber(cid:22)ows
(cid:17)cf(cid:1) Sections (cid:12) and (cid:21) as well as (cid:3)(cid:4)(cid:12)(cid:8)(cid:18)(cid:1) Then the central open problem concerning invariant
measures can be formulated in the following way(cid:1)
All standard examples act on biquotients M of a Lie group G(cid:1) We call a submanifold
(cid:3)
M of M homogeneous if its preimage in G is a coset of a closed subgroup(cid:1) We call a
measure on a (cid:10)nite union of homogeneous submanifolds Haar if its restriction to any of the
homogeneous submanifolds can be constructed by projecting Haar measure on a coset in G
to M(cid:1)
k k k
Main Conjecture(cid:1) Let (cid:0) be a standard Anosov action of Z(cid:2)(cid:1) Z or R (cid:1) k (cid:1) (cid:7) on a
manifold M(cid:0) Then any (cid:0)(cid:2)invariant ergodic Borel probability measure (cid:1) is either Haar
measure on a homogeneous real algebraic subspace or the support of (cid:1) is a homogeneous
(cid:3)
subspace M which (cid:3)bers in an (cid:0)(cid:2)invariant way over a manifold N such that the (cid:0)(cid:2)action
on N reduces to a rank one action(cid:1) i(cid:0)e(cid:0) the action of Z(cid:2)(cid:1) Zor R(cid:0)
In particular(cid:1) if the support of (cid:1) is all of M then (cid:1) is Haar measure on M(cid:0)
The second alternative includes measures supported on closed orbits of the action(cid:1) The
set of such orbits is always dense(cid:1) Aside from those measures(cid:6) the second alternative does
not appear in the standard toral examples and appears to be rather exceptional in the
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:12)
symmetric space examples(cid:1)
Asimilarconjecturecanbestatedforamoregeneralclassofpartiallyhyperbolic actions
where one may have to allow natural measures on some non(cid:2)homogeneous real algebraic
submanifolds(cid:1)
In this paper we consider invariant ergodic measures for certain homogeneous actions
of higher rank abelian groups(cid:1) Our main assumption is similar to that of Rudolph and
Johnson(cid:6) namely thatsome element has positive entropyw(cid:1)r(cid:1)t(cid:1) the measure in question(cid:1) In
k
the most general case(cid:6) we have to assume more(cid:1) For R (cid:2)actions for example(cid:6) it is su(cid:0)cient
k
to assume that every one(cid:2)parameter subgroup is ergodic(cid:1) The similar assumptions for Z(cid:2)
k
andZ(cid:2)(cid:2)actionsarethatoneparametersubgroupsofthesuspension andcorrespondinglythe
suspension of the natural extension are ergodic(cid:1) In particular(cid:6) all mixing measures satisfy
these assumptions(cid:1) These conditions exclude measures coming from Rees(cid:20)s examples since
those measures are not ergodic with respect to certain one(cid:2)parameter subgroups(cid:1)
Under those or slightly weaker assumptions(cid:6) we show in the toral and semisimple (cid:17)sym(cid:2)
metric space(cid:18)cases thatthemeasureis Haarmeasure onahomogeneousalgebraic subspace
(cid:17)Theorems (cid:14)(cid:1)(cid:4) and (cid:5)(cid:1)(cid:4)(cid:26) Corollaries (cid:14)(cid:1)(cid:7) and (cid:14)(cid:1)(cid:12)(cid:6) and analogous statements for the semisim(cid:2)
ple case(cid:18)(cid:1) In many cases(cid:6) where there are no non(cid:2)trivial homogeneous algebraic invariant
subspaces(cid:6) this implies that the measure is Haar measure on the whole space(cid:1) In the case
of twisted Weyl chamber (cid:22)ows(cid:6) to achieve similar conclusions we need to assume in ad(cid:2)
dition that the projection to the semisimple factor has positive entropy for some element
(cid:17)Theorem (cid:5)(cid:1)(cid:7)(cid:18)(cid:1)
For certain toral actions(cid:6) essentially the totally non(cid:2)symplectic actions(cid:6) the extra as(cid:2)
sumptions can be removed(cid:1) Thus we obtain a generalization of the Rudolph and Johnson
results which covers certain commuting expanding toral endomorphisms(cid:6) Anosov actions
n(cid:1)(cid:0) n
of higher rank subgroups of Z on T by automorphisms and many other examples
(cid:17)Corollary (cid:21)(cid:1)(cid:13)(cid:18)(cid:1)
The main idea of our argument is to decompose the invariant measure into conditionals
along stable and unstable foliations of various elements of the action(cid:1) These foliations
are homogeneous(cid:1) By looking at conditionals at various invariant subfoliations we show
that some of those conditional measures are either atomic or Haar along a homogeneous
subfoliation(cid:1) In the (cid:10)rst case(cid:6) the entropy of some and then every element is zero(cid:1) In the
second case(cid:6) rigidity follows in the toral case from unique ergodicity of a linear (cid:22)ow on
the torus on its orbit closures and in the semisimple and twisted cases from M(cid:1) Ratner(cid:20)s
classi(cid:10)cation of invariant measures for homogenous actions of unipotent groups (cid:3)(cid:7)(cid:12)(cid:8)(cid:1)
Let us point out that our method which is based on the local structure of stable and
k
unstable foliations for various elements breaks down for symplectic actions of Z on even(cid:27)
dimensional tori(cid:1) Such actionsmaybe totallyirreducible (cid:17)noinvariantrationalsubtori(cid:18)(cid:26)ex(cid:2)
(cid:1) (cid:3)
plicit examplesofthatkindstartingfromZ actionsonT wereshowntousbyL(cid:1)Vaserstein(cid:1)
However(cid:6)thelocalstructureofsuchactionsaspresentedbyLyapunovdecompositions(cid:6) Weyl
chambers and Lyapunov hyperplanes (cid:17)see next section(cid:18) is undistinguishable from that of
the products of rank one actions(cid:1)
Acknowledgements (cid:1) We would like to thank H(cid:1) Furstenberg for alerting us to some
errors in the original manuscript(cid:1) We would also like to express our deep gratitude to the
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:13)
Mathematical Sciences Research Institute in Berkeley(cid:6) where this paper was written(cid:6) for
providing a stimulating environment(cid:6) ideal working conditions and for (cid:10)nancial support(cid:1)
The second author is also grateful for the hospitality of the Pennsylvania State University
during several visits(cid:1)
(cid:1) Lyapunov exponents
k k k
We will study Anosov and(cid:6) more generally(cid:6) partially hyperbolic actions of Z(cid:2)(cid:6) Z and R (cid:1)
For a general discussion of such actions we refer to (cid:3)(cid:4)(cid:14)(cid:6) (cid:4)(cid:13)(cid:8)(cid:1) As we will see(cid:6) it is more
k
convenient for our approach to operate with R (cid:2)actions(cid:1) Therefore let us (cid:10)rst explain how
k k
to pass from an action of Z to R (cid:1) This is the so(cid:2)called suspension construction(cid:1)
k k k k k
Suppose Z acts on N(cid:1) Embed Z as a lattice in R (cid:1) Let Z act on R (cid:2) N by
z(cid:17)x(cid:2)m(cid:18)(cid:16) (cid:17)x(cid:3)z(cid:2)z(cid:3)m(cid:18) and form the quotient
k k
M (cid:16) R (cid:2)N(cid:4)Z(cid:3)
k k k
Note that the action of R on R (cid:2)N by x(cid:3)(cid:17)y(cid:2)n(cid:18)(cid:16) (cid:17)x(cid:28)y(cid:2)n(cid:18)commutes with the Z(cid:2)action
k k
and therefore descends to M(cid:1) This R (cid:2)action is called the suspension of the Z(cid:2)action(cid:1)
k k
Note that any Z(cid:2)invariant measure on N lifts to a unique R (cid:2)invariant measure on the
suspension(cid:1)
k k
Furthermore we can pass from a Z(cid:2)(cid:2)action to a Z(cid:2)action by a natural projective limit
construction in an appropriate category(cid:1) This construction is explained in detail for toral
endomorphisms in Section (cid:12) where it is called the solenoid construction(cid:1) As we will see in
the appendix(cid:6) the solenoids are locally modeled on the products of certain p(cid:2)adic rings of
k
integers with R (cid:1) Let us also mention that any invariant measure on the torus canonically
lifts to the solenoid(cid:1)
k
A crucial role in our analysis of R (cid:2)actions is played by the Lyapunov exponents(cid:1) Con(cid:2)
k k
sider a measure preserving ergodic action of R on a space X(cid:1) Suppose R acts by bundle
automorphisms on a bundle over X with products of real and p(cid:2)adic vectorspaces as (cid:10)bers
covering the given action on X(cid:1) For a single element a in the group and a vector v in the
extension(cid:6) theLyapunovexponent (cid:5)(cid:17)a(cid:2)v(cid:18)is de(cid:10)ned in theusual way(cid:17)comparewith (cid:3)(cid:7)(cid:7)(cid:6) ch(cid:1)
V(cid:8)(cid:18)(cid:1) There is a decomposition into Lyapunov subspaces of the extension a(cid:1)e(cid:1) such that the
di(cid:9)erent Lyapunov exponents of a are given as Lyapunov exponents of a and some vector
in the Lyapunov space(cid:1) Due to the commutativity of the group(cid:6) we can (cid:10)nd a common
re(cid:10)nement of the Lyapunov decompositions of single elements of the group(cid:1) We will call
this re(cid:10)ned decomposition the Lyapunov decomposition of the extension(cid:1) This allows us to
consider the Lyapunov exponents (cid:5)(cid:17)(cid:2)v(cid:18) for v in a Lyapunov space of the extension as a
real valued functional on the group(cid:1) Since the acting group is abelian(cid:6) the Lyapunov expo(cid:2)
nents are linear functionals on the group(cid:1) A particular example of such an extension for a
smooth system is its derivative(cid:1) We refer to (cid:3)(cid:5)(cid:8) for a more detailed exposition of Lyapunov
exponents in this case(cid:1)
k k
When wespeak aboutLyapunov exponents ofa Z(cid:2)action oraZ(cid:2)(cid:2)action wewill always
mean those for the suspension and correspondingly the suspension of the natural extension
k
of the given action(cid:1) Consider the (cid:10)nitely many hyperplanes in R de(cid:10)ned by the vanishing
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:14)
of the functionals(cid:1) We will call these hyperplanes the Lyapunov hyperplanes(cid:1) Let us call an
k
element a(cid:4) R regular if it does not belong to the kernel of any non(cid:2)trivial functional(cid:1) All
other elements are called singular(cid:1) Call a singular element generic if it belongs to only one
k
Lyapunov hyperplane(cid:1) Note that the tangent space to the R (cid:2)orbit de(cid:10)nes the identically
(cid:15) Lyapunov exponent(cid:1) Let us emphasize that Lyapunov exponents may be proportional
to each other with positive or negative coe(cid:0)cients(cid:1) In this case(cid:6) they de(cid:10)ne the same
Lyapunov hyperplane(cid:1)
k
The Lyapunov hyperplanes divide R into (cid:10)nitely many open connected components(cid:6)
called the Weyl chambers of the action(cid:1) Thus every regular element belongs to a unique
Weyl chamber(cid:1) Every generic singular element belongs to the common boundary of exactly
two Weyl chambers(cid:1) The system of Weyl chambers is symmetric w(cid:1)r(cid:1)t(cid:1) the origin(cid:1) Thus
for any Weyl chamber C(cid:6) (cid:3)C is also a Weyl chamber(cid:1)
k k
Note that the Lyapunov hyperplanes cannot be are not derictly seen from a Z or Z(cid:2)(cid:2)
k
action as the hyperplanes are not rational in general(cid:1) This is one of the reasons making R
actions a more convenient object of study(cid:1)
In all homogeneous examples(cid:6) standard or not(cid:6) the Lyapunov exponents for the deriva(cid:2)
tive extension are de(cid:10)ned and constant everywhere(cid:1) In particular(cid:6) they are independent
of the invariant measure(cid:1) They determine a splitting of the tangent bundle (cid:17)which may
havep(cid:2)adic componentsin thesolenoid case(cid:18)intoinvariant subbundles called theLyapunov
spaces(cid:1) Let us emphasize that the Lyapunov spaces in the p(cid:2)adic directions correspond to
m
closed subgroups of some Zp (cid:1) The dimension of each Lyapunov space will be called the
multiplicity of the exponent (cid:17)where dimension for a p(cid:2)adic direction is the dimension of
corresponding p(cid:2)adic modules(cid:6) c(cid:1)f(cid:1) the Appendix(cid:18)(cid:1) The multiplicity of the (cid:15) exponent is
at least k(cid:1) If the multiplicity of the (cid:15) exponent is exactly k(cid:6) we call the action Anosov(cid:1) A
regular element for an Anosov action is called an Anosov element(cid:1)
k (cid:2) (cid:1)
Foranelement a(cid:4) R let us de(cid:10)ne the stable(cid:1) unstableand neutral distribution Ea (cid:2)Ea
n
and Ea as the sum of the Lyapunov spaces for which the value of the corresponding Lya(cid:2)
punov exponent on a is negative(cid:6) positive and (cid:15) respectively(cid:1) The neutral distribution
k
for any element of an R (cid:2)action contains the tangent distribution to the orbit(cid:26) in the
non(cid:2)Anosov partially hyperbolic case it also contains other directions corresponding to the
Lyapunov exponents identically equal to (cid:15)(cid:1) We will be interested in the complement to
these (cid:23)trivial(cid:24) directions in the neutral distribution of a singular element(cid:1) It is de(cid:10)ned as
follows(cid:1)
Note that the stable and unstable distributions are constant on a Weyl chamber(cid:1) Note
furthermore that the sum of stable and unstable distributions is constant for all regular
H
elements(cid:1) We will denote this sum by E (cid:1) For singular elements the neutral distributions
are bigger than for regular ones(cid:1) For example(cid:6) for a generic singular element the neutral
distribution containsadirectionwhichisstableforoneadjacentWeylchamberandunstable
for the other(cid:1) We will call the intersection of the neutral distribution for an element a
H (cid:4)
with the distribution E the center distribution of a and denote it by Ea(cid:1) The center
distribution for any singular element a always contains directions on which the derivative
of a acts isometrically w(cid:1)r(cid:1)t(cid:1) a canonical homogeneous metric(cid:1) We denote the distribution
(cid:4) I
of isometric directions inside Ea by Ea(cid:1)
Let us note that the Lyapunov spaces in the standard toral examples always integrate
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:21)
to a(cid:0)ne foliations (cid:17)possibly with a p(cid:2)adic part(cid:18)(cid:1) In the symmetric space examples(cid:6) some
of the Lyapunov spaces may not be integrable (cid:17)cf(cid:1) the discussion in the proof of Theo(cid:2)
rem (cid:5)(cid:1)(cid:4)(cid:18)(cid:1) However(cid:6) stable(cid:6) unstable and center distributions as well as their intersections
(cid:17)for di(cid:9)erent elements(cid:18) are always integrable and integrate to homogeneous foliations(cid:1) In
fact(cid:6) the stable and unstable foliations of any element are always the orbit foliations of a
unipotent subgroup(cid:1) For an element a we will denote the integral foliations of the stable(cid:6)
(cid:2) (cid:1) (cid:4) I (cid:2) (cid:1) (cid:4) I
unstable(cid:6) center and isometric distributions Ea (cid:2)Ea(cid:2)Ea and Ea by Wa (cid:2)Wa (cid:2)Wa and Wa(cid:1)
(cid:2) Toral endomorphisms(cid:3) solenoids and their suspensions
k
Consider an embedding (cid:0) of Z(cid:2) into the semigroup of non(cid:2)singular m(cid:2)m integer matri(cid:2)
k m
ces(cid:1) Then Z(cid:2) acts on the torus T by endomorphisms(cid:1) Note that this includes actions
k k
of Z by toral automorphisms by restricting to Z(cid:2)(cid:1) We will always assume that every
k
non(cid:2)trivial element of Z(cid:2) acts ergodically with respect to Haar measure(cid:6) or equivalently(cid:6)
k
that no non(cid:2)trivial element of Z(cid:2) has eigenvalues that are roots of unity(cid:1) Such an action is
called irreducible if no (cid:10)nite cover splits as a product(cid:1) Irreducible actions by ergodic toral
endomorphisms are called standard actions(cid:1) Furthermore(cid:6) in agreement with the terminol(cid:2)
k
ogy of the previous section(cid:6) we will call (cid:0) Anosov if the image of Z(cid:2) contains matrices
without eigenvalues on the unit circle(cid:1)
Note that if the actions admits a factor on which the action reduces to an action of Z
or Z(cid:2)(cid:6) then invariant measures cannot be rigid(cid:1) In this case however(cid:6) any element in the
kernel of the action on the factor has (cid:4) as an eigenvalue(cid:1) Thus actions by ergodic toral
automorphismsdonotadmitsuchfactors(cid:1) Conjecturally(cid:6) thepresence of(cid:23)rankonefactors(cid:24)
is the only obstruction to rigidity(cid:1)
(cid:0) k
To make these actions invertible we will introduce the natural extension (cid:0) (cid:29) Z (cid:5)
Aut(cid:17)S(cid:18) of (cid:0) where S is the solenoid obtained from the torus as follows(cid:1)
k
Let A(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)Ak be the images of the generators of Z(cid:2)(cid:1) Then we get a projective system
H
Hj
(cid:1)
(cid:1) m
(cid:1)(cid:0)(cid:1)T HA(cid:0)
(cid:0) (cid:1) (cid:1) Hj
(cid:1) (cid:1) m
H (cid:1) (cid:1) (cid:0)(cid:1)T
Hj (cid:0)
(cid:1) m Ak
(cid:1)(cid:1)(cid:0)(cid:1)T
(cid:0)
where the maps are given by the Ai(cid:1) We let the solenoid S be the projective limit of this
system in the category of compact topological groups(cid:1)
k
m Z
The solenoid can be realized as a subset of (cid:17)T (cid:18) as follows(cid:1) Let (cid:6)i be the i(cid:20)th shift
k
on Z i(cid:1)e(cid:1) (cid:6)i(cid:17)j(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)ji(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)jk(cid:18)(cid:16) (cid:17)j(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)ji(cid:28)(cid:4)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)jk(cid:18)(cid:1) Then set
k
m Z
S (cid:16) f(cid:7) (cid:4) (cid:17)T (cid:18) j (cid:7)(cid:0)ij (cid:16) Ai(cid:7)jg(cid:3)
k
m Z
The solenoid is a compact subgroup of (cid:17)T (cid:18) with the product topology(cid:1) Its dual
m m
is a subgroup of Q (cid:6) more precisely it is contained in (cid:17)Z(cid:17)p(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)pl(cid:18)(cid:18) where p(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)pl
are those prime integers which occur in the prime decomposition of the determinant of at
least one of the matrices A(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)Ak and Z(cid:17)p(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)pl(cid:18) is the subgroup of rational numbers
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:5)
k
whose denominators are only divisible by p(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)pl(cid:1) Note that Z acts on S naturally by
(cid:0) m
coordinate shifts(cid:1) Let us denote this action by (cid:0) (cid:1) The solenoid is a (cid:10)bration over T
with Cantor set (cid:10)bers by mapping (cid:7) (cid:4) S to (cid:7)(cid:17)(cid:15)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)(cid:15)(cid:18)(cid:1) The projection intertwines the
(cid:0) k
(cid:0) (cid:2)action restricted to Z(cid:2) with (cid:0)(cid:1) Note that a local cross(cid:2)section to this (cid:10)bration is given
by the local connected component of (cid:7)(cid:1) We will call this transversal the toral direction at
(cid:7)(cid:1) Note that the projection is one(cid:2)to(cid:2)one if and only if all Ai are invertible(cid:1)
(cid:0)
Every (cid:0)(cid:2)invariant measure lifts in a unique fashion to an (cid:0) (cid:2)invariant measure on S(cid:1)
There is a natural H(cid:11)older structure on the solenoid which comes from any metric on
the product of the form
(cid:3) dTm(cid:17)(cid:7)j(cid:2)(cid:7)j(cid:3)(cid:18)
dc(cid:17)(cid:7)(cid:2)(cid:7)(cid:18)(cid:16) kjk
c
j
X
where where c (cid:8) (cid:4) and dTm is the standard metric on the torus(cid:1) Note that the Ho(cid:11)lder
structure is independent of c(cid:1) This also allows us to de(cid:10)ne exponential convergence along
(cid:0)
the (cid:10)ber and hence stable(cid:6) unstable and neutral spaces for the elements of (cid:0) (cid:1)
Formostofthepaper(cid:6)andinparticularfortheMainTheorem(cid:14)(cid:1)(cid:4)(cid:6)itissu(cid:0)cienttohave
these rough dynamical structures(cid:1) For certain applications in Section (cid:21) however(cid:6) we need
to de(cid:10)ne speci(cid:10)c exponential speeds of expansion and contraction(cid:6) in the p(cid:2)adic directions
i(cid:1)e(cid:1) Lyapunov exponents(cid:1) To do this(cid:6) we need a more subtle metric structure on S which
requires an alternative(cid:6) more arithmetic description of the solenoid(cid:1) Its main advantage is
thatwecancanonically de(cid:10)ne aspecial metricdonS which givesaLipschitz structureonS
and de(cid:10)nes Lyapunov exponents on S which agree with the standard Lyapunov exponents
in the toral direction(cid:1) The metric d is Ho(cid:11)lder equivalent to dc(cid:1) Since these issues are
irrelevant to the invertible case and the Main Theorem(cid:6) we only give this description in an
appendix(cid:1)
(cid:4) Conditional measures and entropy
Let us brie(cid:22)y recall how a probability measure (cid:9) on M determines a system of conditional
measures on a foliation F(cid:1) Denote by B the Borel (cid:6)(cid:2)algebra on M(cid:1) A measurable partition
(cid:10) of M is a partition of M such that(cid:6) up to a set of measure (cid:15)(cid:6) the quotient space M(cid:4)(cid:10)
is separated by a countable number of measurable sets (cid:3)(cid:7)(cid:14)(cid:8)(cid:1) For every x in a set of full
(cid:1)
(cid:9)(cid:2)measure there is a probability measure (cid:9)x de(cid:10)ned on (cid:10)(cid:17)x(cid:18)(cid:6) the element of (cid:10) containing x(cid:6)
and satisfying the following properties(cid:29) If B(cid:1) is the sub(cid:2)(cid:6)(cid:2)algebra of B whose elements are
(cid:1)
unions of elements of (cid:10)(cid:6) and A (cid:6)M is a measurable set(cid:6) then x (cid:7)(cid:5) (cid:9)x(cid:17)A(cid:18) is B(cid:1)(cid:2)measurable
(cid:1) (cid:1)
and (cid:9)(cid:17)A(cid:18)(cid:16) (cid:9)x(cid:17)A(cid:18)(cid:9)(cid:17)dx(cid:18)(cid:1) These conditions determine the measures (cid:9)x uniquely(cid:1)
Given a continuous foliation F(cid:6) let F(cid:17)x(cid:18) denote the leaf through x(cid:1) The partition
R
into the leaves of F is not a measurable partition in general(cid:1) (cid:17)Although the point of the
Proposition below as well as of the most of the arguments in Section (cid:14) is that in the zero
entropy situation they in fact are measurable(cid:1)(cid:18) Let (cid:6)(cid:17)F(cid:18) denote the (cid:6)(cid:2)algebra of all sets
that consist a(cid:1)e(cid:1)(cid:1) of complete leaves of F(cid:1) It corresponds to a unique measurable partition
which is called the measurable hull of F(cid:6) and is denoted by (cid:10)(cid:17)F(cid:18)(cid:1) It is the (cid:10)nest measurable
partition whose elements consist a(cid:1)e(cid:1) from the entire leaves of F(cid:1) Unless it is trivial(cid:6) it
is usually hard to describe geometrically(cid:1) We will be primarily interested in the intergal
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:19)
foliations of various distributions described in Section (cid:7)(cid:1) Conditional measures on leaves
F
of such a foliation are (cid:6)(cid:2)(cid:10)nite locally (cid:10)nite measures (cid:9)x de(cid:10)ned up to a multiplicative
constant(cid:1) In other words(cid:6) for almost every x (cid:4) M and for open sets A(cid:2)B (cid:6) F(cid:17)x(cid:18) with
F
(cid:2)x(cid:5)A(cid:6)
compact closures one can canonically de(cid:10)ne the ratio (cid:2)xF(cid:5)B(cid:6)(cid:1)
In the homogenous case in question as well as in some other cases this can be done as
follows(cid:1) Take a small homogenous transversal T to F(cid:17)x(cid:18) at x and translate it to cover a
neighborhood of large enough disc D in F(cid:17)x(cid:18) which contains both A and B(cid:1) Thus in this
neighborhood we have a product structure modeled on D(cid:2)T(cid:1) There is also a metric which
is translation invariant(cid:1) Let T(cid:17)(cid:11)(cid:18)(cid:6) T be the (cid:11) ball around x(cid:1) Then
F
(cid:9)x(cid:17)A(cid:18) (cid:9)(cid:17)A(cid:2)T(cid:17)(cid:11)(cid:18)(cid:18)
F (cid:16) lim (cid:3)
(cid:9)x(cid:17)B(cid:18) (cid:3)(cid:4)(cid:4) (cid:9)(cid:17)B (cid:2)T(cid:17)(cid:11)(cid:18)(cid:18)
There is an alternative way of describing conditional measures which works in a more
general situation(cid:1) Call a measurable partition (cid:10) subordinate to F if for (cid:9)(cid:2)a(cid:1)e(cid:1) x we have
(cid:10)(cid:17)x(cid:18) (cid:6) F(cid:17)x(cid:18) and (cid:10)(cid:17)x(cid:18) contains a neighborhood of x open in the submanifold topology
of F(cid:17)x(cid:18)(cid:1) Note that two di(cid:9)erent partitions subordinate to the same foliation determine
conditional measures that are scalar multiples when restricted to the intersection of an
element of one partition with an element of the other partition(cid:1) Thus there is a locally
F
(cid:10)nite measure (cid:9)x on F(cid:17)x(cid:18) uniquely de(cid:10)ned up to scaling that restricts to a scalar multiple
F
of a conditional measure for each partition subordinate to F(cid:1) The measures (cid:9)x form the
system of conditional measures on the leaves of F(cid:1) In more general situations which do
not concern us in this paper a certain care is needed to justify the fact that conditional
measures are really correctly de(cid:10)ned up to a constant scalar multiple(cid:1) However in order
to show connections between trivialization of conditional measures it is enough to see that
conditional measures are de(cid:10)ned up to a scalar function which is of course quite obvious
from the preceeding construction in a fairly great generality(cid:1) Of course(cid:6) at the end this is
not surprising at all since the conclusion will be that the conditional measures are atomic(cid:6)
hence (cid:10)nite and can be normalised so that the partition into leaves is measurable(cid:30)
k
Given a (cid:4) R and an a(cid:2)invariant measure (cid:1)(cid:6) we denote the partition into ergodic
components of (cid:1) under a by (cid:10)a(cid:1)
Let us recall the relation between conditional measures and entropy(cid:1) It is well(cid:2)known
that entropy is related to exponential contraction and expansion(cid:1) In order to accomodate
solenoids(cid:6) we will formulate a criterion for the vanishing of metric entropy in the context
of foliated compact metric spaces(cid:1)
n
The underlying spaces forour actions are locally isometric with the product of some R
mp
with (cid:10)nitely many Qp (cid:1) All the invariant distributions and associated foliations are also
locally isometric to such products(cid:1) Recall that the box dimension of a metric space (cid:17)M(cid:2)d(cid:18)
is given by
log(cid:17)Nd(cid:17)(cid:12)(cid:18)(cid:18)
limsup(cid:3)
(cid:4)(cid:4)(cid:4) log(cid:17)(cid:12)(cid:18)
whereNd(cid:17)(cid:12)(cid:18)isthemaximumnumberofdisjoint (cid:12)(cid:2)balls inM(cid:1) Let(cid:12)(cid:8) (cid:13) (cid:8) (cid:15)andletNd(cid:17)(cid:12)(cid:2)(cid:13)(cid:18)
be the maximum number of disjoint (cid:13)(cid:2)balls in any (cid:12)(cid:2)ball(cid:1) there is a constant D (cid:8) (cid:15) such
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:25)
that for all small (cid:12) (cid:8) (cid:13) (cid:8) (cid:15)(cid:6)
logNd(cid:17)(cid:12)(cid:2)(cid:13)(cid:18)
(cid:4) (cid:14) D(cid:3)
log(cid:17)(cid:5)(cid:18)
Let (cid:17)F(cid:2)dF(cid:18) and (cid:17)T(cid:2)dT(cid:18) be metric spaces(cid:1) A foliation F of a metric space (cid:17)M(cid:2)d(cid:18)(cid:6)
modeled on F with transversal T is adisjoint decomposition ofM intosubspaces Fx(cid:6) called
the leaves of F such that each Fx is the Lipschitz image of F and for every point x (cid:4) X(cid:6)
there is a neighborhood U such that U is bi(cid:2)Lipschitz with the metric product UF (cid:2)UT
where Uf and UT are neighborhoods in F and T respectively(cid:6) and where the bi(cid:2)Lipschitz
map takes UF (cid:2)ftg for all t (cid:4) UT to the intersection of a leaf of F with U(cid:1) We say that
a pair of foliations F and G of M de(cid:10)ne a local product structure if F is modelled on F
with transversal T and G is modelled on T with transversal F and the bi(cid:2)Lipschitz maps
de(cid:10)ned locally respect both foliations simultaneously(cid:1)
Proposition (cid:2)(cid:3)(cid:4) Let M be a compact metric space of (cid:3)nite local box dimension(cid:0) Suppose
F and G are foliations on M that de(cid:3)ne a local product structure(cid:0) Let (cid:15) (cid:29) M (cid:5) M be
a bi(cid:2)Lipschitz homeomorphism preserving F and G(cid:1) which locally strictly contracts F and
such that for every (cid:12) (cid:8) (cid:15) there is a C(cid:4) (cid:8) (cid:15) such that for all n (cid:1) (cid:15) and y (cid:4) G(cid:17)x(cid:18) the
(cid:1)n (cid:1)n (cid:4)n
distance d(cid:17)(cid:15) (cid:17)y(cid:18)(cid:2)(cid:15) (cid:17)x(cid:18)(cid:18)(cid:8) C(cid:4)e d(cid:17)x(cid:2)y(cid:18) if d(cid:17)x(cid:2)y(cid:18)(cid:14) (cid:12)(cid:0)
F
Then(cid:1) if (cid:1) is a Borel probability measure on M(cid:1) and (cid:1)x its system of conditional mea(cid:2)
F F
sures(cid:1) the metric entropy h(cid:6) (cid:16) (cid:15) if and only if for (cid:1)(cid:2)a(cid:0)e(cid:0) x(cid:1) (cid:1)x is atomic(cid:0) In this case(cid:1) (cid:1)x
is supported on a single point(cid:0)
In this paper(cid:6) we will only need this statement in one direction(cid:6) namely that the metric
F
entropy is (cid:15) if the conditional measures (cid:1)x are atomic(cid:1) We will describe the proof of this
F
direction(cid:1) First(cid:6) note that the conditional measure (cid:1) is supported on a single point if it is
atomic(cid:1) Indeed(cid:6) if x is an atom of the conditional measure(cid:6) there is a small neighborhood
F F F
U of x in the leaf such that (cid:1)x(cid:17)U (cid:3) fxg(cid:18) (cid:14) (cid:12)(cid:1)x(cid:17)fxg(cid:18)(cid:1) Pushing (cid:1)x backward and using
F
Poincar(cid:31)e recurrence(cid:6) we see that for a typical x(cid:6) (cid:1)x is concentrated at x(cid:1)
F
Now assume that (cid:1)x is supported in a single point(cid:1) Then we can (cid:10)nd a set of full
measure which intersects every F(cid:2)leaf in at most one point(cid:1) In particular(cid:6) the intersection
of this set with a neighborhood with local product structure is the graph of a measurable
function de(cid:10)ned on an open set U (cid:6) T with values in an open set V (cid:6) F(cid:1) By Lusin(cid:20)s
theorem(cid:6) there is a compact set K of arbitrarily large measure which is a (cid:10)nite union of
graphs of continuous maps from subsets of T to F(cid:1) Let L be a Lipschitz constant for (cid:15)(cid:1)
n
Pick an n and (cid:13) (cid:8) (cid:15) such that L (cid:13) is small(cid:1) Consider a partition (cid:10) of M with two types
of elements(cid:29) intersections of sets of diameter less than (cid:13) with K and with M nK(cid:1) It is
(cid:0) (cid:1)n
well known that h(cid:17)(cid:15)(cid:2)(cid:10)(cid:18) (cid:8) n H(cid:17)(cid:10)(cid:2)(cid:15) (cid:10)(cid:18)(cid:1) The latter quantity is estimated separately for
(cid:1)n
the preimages under (cid:15) of the two types of elements of (cid:10)(cid:1) In both cases(cid:6) we just estimate
(cid:1)n
the contribution of each element c (cid:4) (cid:15) (cid:10) by the number of elements of (cid:10) which have non(cid:2)
(cid:1)n (cid:4)n
empty intersections with c(cid:1) Forc in (cid:15) K(cid:6) weestimate the diameter ofc by C(cid:4)e (cid:13)(cid:6) using
the assumption of the proposition(cid:1) Since the local box dimension is (cid:10)nite(cid:6) the number of
non(cid:2)trivial intersections grows at most exponentially in n with arbitrarily small exponent(cid:1)
(cid:1)n
For (cid:15) (cid:17)c(cid:18) for c (cid:4) (cid:10) where c (cid:6) M nK(cid:6) we have a uniform exponential estimate of the
size and hence number of nontrivial intersections using the Lipschitz constant of (cid:15)(cid:1) Since
February (cid:0)(cid:1) (cid:2)(cid:0)(cid:0)(cid:3) (cid:4)(cid:15)
the measure ofMnK is small(cid:6) the contribution of such elements to the conditional entropy
is small(cid:1)
(cid:5) The main theorem for toral endomorphisms
In all standard toral examples(cid:6) let us consider the tangent bundle to the phase space and
de(cid:10)ne the derivative action(cid:1) (cid:17)In the solenoid case(cid:6) this will include some non(cid:2)Archimedean
components(cid:6) as explained in the Appendix(cid:1) All the arguments in this section however only
usetheArchimedean directions(cid:1) Thuswemayjustconsidertherealtangentbundle overthe
solenoidinthissection(cid:1)(cid:18) Thederivativeisalinearextensionoftheaction(cid:6)andtheLyapunov
exponents are given by logarithms of the appropriate valuations of the eigenvalues(cid:1) By
commutativitywecan(cid:10)ndajointsplittingintosubspacesonwhichtheLyapunovexponents
are constant for each element(cid:1) This is the decomposition into Lyapunov spaces described
in Section (cid:7)(cid:1)
Let us recall again that there is a one(cid:2)to(cid:2)one correspondence between Borel probability
k
ergodic invariant measures for an action of Z(cid:2) by toral endomorphisms and those for the
k
R (cid:2)action which is the suspension ofthe solenoid extension of the toralaction(cid:1) Since in our
k
arguments we will be dealing mostly with R (cid:2)actions obtained as suspensions of solenoid
k
extensions we will adopt the following notation(cid:1) If (cid:1) is an invariant measure for an R (cid:2)
action then (cid:1)Tm will denote the corresponding measure for the toral action(cid:1) Obviously(cid:6)
k
every element of the R (cid:2)action has zero entropy w(cid:1)r(cid:1)t(cid:1) (cid:1) if and only if every element of the
corresponding Zk(cid:2)(cid:2)action has zero entropy w(cid:1)r(cid:1)t(cid:1) (cid:1)Tm(cid:1)
The following theorem is our principal technical result in the toral case(cid:1)
k
Theorem (cid:5)(cid:3)(cid:4) Let (cid:0) be a R (cid:2)action with k (cid:1) (cid:7) induced from a standard action by toral
endomorphisms(cid:0) Assume that (cid:1) is an ergodic invariant measure for (cid:0) such that there
k (cid:2)
are generic singular elements a(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)ak and a regular element b (cid:4) R with Eb totally
Archimedean such that
(cid:2) (cid:4) (cid:2)
(cid:17)(cid:9)(cid:18) Eb (cid:16) (cid:17)Eai (cid:10)Eb (cid:18)
i
X
(cid:4)where the sum need not be direct(cid:5) and such that
(cid:4) (cid:2)
(cid:17)(cid:9)(cid:9)(cid:18) (cid:10)ai (cid:8) (cid:10)(cid:17)Eai (cid:10)Eb (cid:18)(cid:3)
Theneither (cid:1)Tm isHaar measureona rationalsubtorus or every elementof(cid:0) has (cid:6) entropy
w(cid:0)r(cid:0)t(cid:0) (cid:1)(cid:0)
The genericity of the ai is not actually needed as one can easily see from the proof of
the theorem(cid:1)
Remark (cid:1) This theorem generalizes to suspensions of groups of solenoid automorphisms
more general than those obtained from extensions of groups of toral endomorphisms (cid:17)cf(cid:1)
Example (cid:12)(cid:1)(cid:21)(cid:18)(cid:1) The principal di(cid:9)erence in the formulation is that the stable distribution
(cid:2)
Eb is not assumed to be totally Archimedean(cid:1) Conditions (cid:17) (cid:18) and (cid:17) (cid:18) remain the same(cid:1)
The di(cid:9)erences in the proof are not very signi(cid:10)cant(cid:6) and will be left to the reader(cid:1)