A NOTE ON SMOOTH DYNAMICAL SYSTEMS WITH DISCRETE SPECTRUM

ALEX FURMAN

Abstract. We consider the problem of deciding whether a given smooth volume preserving action of a countable group on a compact manifold has a discrete spectrum. We give a sufficient condition in terms of integrability of the norm of the derivative cocycle of the action. The short proof is based on the notion of entropy of a sequence of m.p. transformations, and an estimate of such entropy by the logarithm of the L^{1}-norm of the derivative cocycle.

1. Introduction and statement of results.

An invertible ergodic transformation T of a probability space (X, μ) is said to have discrete spectrum if the space L_{0}^{2}(X, m) is spanned by the eigenfunctions of the corresponding unitary operator. By the classical result of von Neumann and Halmos [vNH] transformations with discrete spectrum are measure-theoretically isomorphic to a translation T : k ↦ α + k on a compact abelian group K.

It is possible to generalize the notion of discrete spectrum to ergodic actions of a general countable group G on a probability space (X, μ). In this general case eigenfunctions are replaced by finite dimensional invariant spaces, so a G-action has discrete spectrum if L^{2}(X, μ) = ˆ⊕V_{i}, where V_{i} are finite dimensional G-invariant subspaces. In this case Mackey [Ma] has shown that the action can be represented as a translation on a quotient of a compact group. More precisely: there exist compact groups K_{0} ⊆ K and a homomorphism α : G → K with dense image, s.t. the G-action on K/K_{0}, given by g · (kK_{0}) = (α(g)k)K_{0}, is measurably isomorphic to the G-action on (X, μ).

In as much as the measure-theoretic aspects of actions with discrete spectrum are fully understood by the above, the theory of smooth volume preserving diffeomorphisms of compact Riemannian manifolds with discrete spectrum is far from being comprehensive. The purpose of this note is to give a sufficient condition (in terms of the derivative cocycle of the action) on a volume preserving action of a countable group to have a discrete spectrum.

More precisely, we have

Theorem 1. Let G be a countable group acting by volume preserving diffeomorphisms on a compact d-dimensional Riemann manifold M and let D : G × M → GL(d, ℝ) denote the derivative cocycle D(g, x) : T_{x}M → T_{gx}M. Suppose that ∥D(g, x)∥ is bounded in L^{1}(M, ω) uniformly on G, i.e.

sup_{g∈G} ∫_{M} ∥D(g, x)∥ dω(x) < ∞. (1)

Then the G-action on the probability space (M, ω) has discrete spectrum.

Corollary 2. Let G be a countable group of volume preserving diffeomorphisms on a compact Riemann d-dimensional manifold M. Suppose that G preserves a measurable Riemann structure, which is L^{2}(M, ω)-equivalent to the given Riemann structure, i.e. there exists a measurable function B : M → GL(d, ℝ) with ∥B(x)∥, ∥B(x)^{−1}∥ ∈ L^{2}(M, ω), such that the cocycle

C(g, x) = B(gx)^{−1} D(g, x) B(x) (2)

takes values in the compact group O(d). Then the G-action on (M, ω) has discrete spectrum.

Remarks 3. 1. Consider the case G = ℤ. Then an existence of an invariant measurable Riemannian structure is neither sufficient nor necessary for the discrete spectrum property. An example of weakly mixing diffeomorphism preserving some measurable Riemannian structure was constructed by A. Katok [Ka1].

2. Consider the case where G is a discrete group satisfying Kazhdan’s property T. It was proved by R. Zimmer [Zi1] (see also [Fu]) that zero-entropy smooth actions of higher rank lattices (and discrete Kazhdan groups in general) admit a measurable invariant Riemannian structure, and in particular discrete spectrum property guarantees an existence of a measurable invariant Riemannian structure.

3. In [Zi2] R. Zimmer had shown that for smooth actions of discrete Kazhdan groups the existence of an invariant measurable Riemannian structure implies the discrete spectrum property.

The motivation for this note was to give an alternative and easy proof of the latter implication, using the following

Theorem 4 (Zimmer). Let a discrete group G act on a probability space (X, μ) so that the G-representation on L_{0}^{2}(X, μ) does not contain weakly the trivial representation 1_{G} (for instance: any ergodic action of a Kazhdan group G). Assume that a measurable cocycle D : G × X → GL_{d}(ℝ) is measurably conjugate by B : X → GL_{d}(ℝ) to a compact valued cocycle

C(g, x) = B(gx)^{−1} D(g, x) B(x) : G × X → O(d).

Then for some p > 0, one has ∥B(x)∥, ∥B(x)^{−1}∥ ∈ L^{p}(X, μ), and therefore

sup_{g∈G} ∫_{X} ∥D(g, x)∥^{p/2} dμ(x) < ∞. (3)

Sketch of the proof. Let S ⊆ G be a finite set of generators and let a = max_{g∈S} ∥D(g, x)∥_{∞}. For some fixed large enough M, the set E = {x ∈ X | ∥B(x)∥ ≤ M} has positive measure. Now observe that the relation ∥B(gx)∥ = ∥D(g, x) B(x) C(g, x)^{−1}∥ ≤ ∥D(g, x)∥ ∥B(x)∥ implies that ∥B(x)∥ ≤ M a^{n} for all x ∈ E_{n} = ⋃_{g∈S^{n}} g E. The assumption that the Grepresentation on L_{)}^{2}(X, μ) does not contain weakly the trivial one, implies (e.g. [Zi1]) that μ(E_{n}) ≥ 1 − λ^{n} for some λ < 1, depending on the G-representation L_{0}^{2}(X, μ). This gives (3) with p > − log_{a} λ.

Unfortunately we were not able to derive Zimmer’s result [Zi2] in full (invariant measurable Riemannian structure implies discrete spectrum for actions of Kazhdan groups) using the above Theorem and Corollary 2 without additional assumptions on the action, which would guarantee p ≥ 2 in (3). Yet Theorem 4 and Corollary 2 suffice to show that if ∥D(g, x)∥ < 1 + ϵ for all g in a finite generating set G_{0} for some ϵ > 0 depending only on Kazhdan constants of G, and G admits an invariant measurable Riemann structure, then the actions has discrete spectrum.

Theorem 1 is an application of the notion of sequential entropy, which was introduced by Kushnirenko in [Ku1], [Ku2]. We use a slight modification of the original definition:

Definition 5. Given a sequence (g_{1}, g_{2}, …) ∈ G^{ℕ} of measure preserving transformations of a probability space (X, μ) and finite measurable partition P of X, define the quantities

\overline{h}(g_{1}, g_{2}, …, P) = lim sup_{n→∞} \frac{1}{n} H(⋁_{i=1}^{n} g_{i}^{−1}P) \underline{h}(g_{1}, g_{2}, …, P) = lim inf_{n→∞} \frac{1}{n} H(⋁_{i=1}^{n} g_{i}^{−1}P).

The upper and the lower sequential entropies with respect to the sequence of transformations (g_{1}, g_{2}, …) are defined as

\overline{h}(g_{1}, g_{2}, …) = sup_{P} \overline{h}(g_{1}, g_{2}, …, P), \underline{h}(g_{1}, g_{2}, …) = sup_{P} \underline{h}(g_{1}, g_{2}, …, P)

where the sup is taken over all finite partitions P of X.

Theorem 1 is a direct consequence of the following Theorems 6 and 7, which may have an independent interest. In particular, Theorem 6 can be considered as an analog of Margulis-Ruelle inequality for an arbitrary sequence of diffeomorphisms:

Theorem 6. Let a group G act by diffeomorphisms on a compact d-dimensional Riemann manifold M, preserving the volume ω. Then for any sequence {g_{1}, g_{2}, … } in G:

(4) \underline{h}(g_{1}, g_{2}, …) ≤ d · lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x))

The next theorem is a variant of the one given by Kushnirenko [Ku1], where the assumption \overline{h} = 0 is replaced by a weaker one, namely \underline{h} = 0.

Theorem 7. Let G act by measure preserving transformations on a probability space (X, μ). Suppose that \underline{h}(g_{1}, g_{2}, …) = 0 for any sequence (g_{1}, g_{2}, …) in G. Then the G-action on (X, μ) has discrete spectrum.

2. Preliminaries.

Lemma 8 (Kushnirenko [Ku1]). Let M be a compact d-dimensional Riemann manifold. Then there exists a constant C_{1}, s.t. for any sufficiently fine finite partition P of M with a piecewise smooth boundary ∂P

H(P) ≤ d · log ν(∂P) + log C_{1},

where ν denotes the (d − 1) dimensional volume on X restricted to ∂P, and the entropy H is computed with respect to the volume on M.

Sketch of the Proof. One first proves the following inequality: for any positive p_{1}, …, p_{k} with p_{1} + ⋯ + p_{k} = 1, and any d ≥ 2:

H(p_{1}, …, p_{k}) = ∑ −p_{i} log p_{i} ≤ d log (∑ p_{i}^{(d−1)/d})

and taking d = dim M, combines it with the isoperimetric inequality on M:

p_{i}^{(d−1)/d} = (volP^{(i)})^{(d−1)/d} ≤ C_{1}ν(∂P^{(i)})

to derive the result.

Now consider the space of all finite measurable partitions of a general Lebesgue probability space (X, μ), naturally endowed with Rohlin’s metric: ρ(P, Q) = H(P | Q) + H(Q | P). Let us say that a sequence {P_{n}} of finite partitions of a probability space (X, μ) is generating if H(P | P_{n}) → 0 for any finite partition P. One easily verifies the following

Lemma 9. If {P_{n}} is a generating sequence of finite partitions, then for any sequence g_{1}, g_{2}, … of measure preserving transformations:

\underline{h}(g_{1}, g_{2}, …) = sup_{n} \underline{h}(g_{1}, g_{2}, …, P_{n}) and \overline{h}(g_{1}, g_{2}, …) = sup_{n} \overline{h}(g_{1}, g_{2}, …, P_{n}).

Moreover, if ρ(P_{n}, Q_{n}) → 0, then {P_{n}} is generating iff {Q_{n}} is generating.

Given a finite measurable partition P let us denote by V_{P} the finite dimensional subspace of L^{2}(X, μ) functions, measurable with respect to P. For the proof of Theorem 7 we shall need the following

Lemma 10. Fix some finite partition P. Then for any ϵ > 0, there exist δ = δ(P, ϵ) > 0 and η = η(P, ϵ) > 0, s.t. for any partition Q:

1. H(P | Q) ≤ δ implies dist(u, V_{Q}) ≤ ϵ · ∥u∥ for all u ∈ V_{P}.

2. If dist(u, V_{Q}) ≤ η · ∥u∥ for all u ∈ V_{P}, then H(P | Q) ≤ ϵ.

Proof. It is well known that H(P | Q) is small iff the atoms P^{(i)} of P are well approximated (in the L^{1}(X, m)-sense) by Q-measurable sets. More precisely, for a fixed finite partition P and any ϵ > 0 there exist δ′ > 0 and η′ > 0, s.t.

1. H(P | Q) ≤ δ′ implies that for each P^{(i)} ∈ P there exists a Q-measurable set Q_{i} with μ(P^{(i)}△Q_{i}) < ϵ.

2. If for each P^{(i)} ∈ P there exists a Q-measurable set Q_{i} with μ(P^{(i)}△Q_{i}) < η′, then

H(P | Q) ≤ ϵ. Replacing L^{1}(X, m) approximation of a finite number of sets P^{(i)} by an L^{2}(X, m)-approximation proves the lemma.

Lemma 11. For any finite dimensional subspace V ⊂ L^{2}(X, μ) and any ϵ > 0 there exists N = N(ϵ, dim V), s.t. at most N vectors in {g_{i} u} are ϵ-close to V.

Proof. The unit ball in V is compact, denote by N = N(ϵ, dim V) the cardinality of its maximal ϵ/3-separated subset. Let v_{i} be the orthogonal projections of g_{i} u on V. For any choice of (N + 1) vectors in {g_{i} u}, which are ϵ/3-close to V, one can find a pair with ∥v_{j} −v_{k}∥ < ϵ/3 and thus ∥g_{j} u−g_{k} u∥ < ϵ/3+ϵ/3+ϵ/3 = ϵ, contradicting the assumption.

3. Proof of Theorem 6.

Let M is a compact d-dimensional Riemann manifold and let G act by diffeomorphisms on X, preserving the volume ω on M.

Since piecewise smooth partitions generate (M, ω), by Lemma 9 it is enough to prove that given a partition Q of X with a piecewise smooth boundary ∂Q and an ϵ > 0, there exists a finite measurable partition P with ρ(P, Q) < ϵ satisfying

(5) \underline{h}(g_{1}, g_{2}, …, P) ≤ d · lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x))

Fix some partition Q with a piecewise smooth boundary ∂Q and an ϵ > 0. Perturbing Q along some flow which is transversal to ∂Q, one obtains a family of partitions Q_{t}, indexed by 0 ≤ t ≤ 1, each of Q_{t} having a piecewise smooth boundary ∂Q_{t}, so that the following properties hold

• ρ(Q_{t}, Q) < ϵ for each 0 ≤ t ≤ 1

• there exists a constant C (one can take C = O(ϵ^{−1})), so that dν_{t} dt ≤ C dμ, where ν_{t}

denotes the (d − 1) volume on ∂Q_{t}. Partition P with the required properties (5) will be chosen among Q_{t}.

Following Kushnirenko [Ku2], we apply Lemma 8 to show that for a fixed 0 ≤ t ≤ 1 and any n ≥ 1:

\frac{1}{n} H(g_{n}^{−1}Q_{t} ∨ ⋯ ∨ g_{1}^{−1}Q_{t}) ≤ d \frac{1}{n} log ν(∂(g_{n}^{−1}Q_{t} ∨ ⋯ ∨ g_{1}^{−1}Q_{t})) + \frac{1}{n} log C_{1}
≤ d \frac{1}{n} log (∑_{k=1}^{n} ν(g_{k}^{−1}∂Q_{t})) + O(\frac{1}{n})

≤ d \frac{1}{n} log (∑_{k=1}^{n} ∫_{Q_{t}} ∥ ∧^{d−1} D(g_{k}^{−1}, x)∥ dν_{t}(x)) + O(\frac{1}{n})

Since ∥ ∧^{d−1} D(g_{n}^{−1}, x)∥ = det D(g_{n}^{−1}, x) · ∥D(g_{n}^{−1}, x)^{−1}∥ = ∥D(g_{n}, g_{n}^{−1}x)∥, we can write

\frac{1}{n} H(g_{n}^{−1}Q_{t} ∨ ⋯ ∨ g_{1}^{−1}Q_{t}) ≤ d \frac{1}{n} log (∑_{k=1}^{n} ∫_{Q_{t}} ∥D(g_{k}, g_{k}^{−1}x)∥ dν_{t}(x)) + O(\frac{1}{n}) (6)

Consider a sequence of functions ϕ_{n} : [0, 1] → [0, ∞) defined by

ϕ_{n}(t) = ∑_{k=1}^{n} ∫_{Q_{t}} ∥D(g_{k}, g_{k}^{−1}m)∥ dν_{t}(m)

Since ϕ_{n} are non-negative, Fatou’s lemma yields:

∫_{0}^{1} lim inf_{n→∞} \frac{1}{n} log ϕ_{n}(t) dt ≤ lim inf_{n→∞} \frac{1}{n} ∫_{0}^{1} log ϕ_{n}(t) dt

by concavity of log():

≤ lim inf_{n→∞} \frac{1}{n} log ∫_{0}^{1} ϕ_{n}(t) dt = lim inf_{n→∞} \frac{1}{n} log (∫_{0}^{1} ∫_{Q_{t}} ∑_{k=1}^{n} ∥D(g_{k}, g_{k}^{−1}x)∥ dν_{t}(x) dt)

and, using the property dν_{t} dt ≤ C dω and g_{k}^{−1}ω = ω, we get

≤ lim_{n→∞} \frac{1}{n} log (∑_{k=1}^{n} ∫_{M} ∥D(g_{k}, g_{k}^{−1}x)∥ dω(x)) = lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x))

Therefore

(7) ∫_{0}^{1} lim inf_{n→∞} \frac{1}{n} log ϕ_{n}(t) dt ≤ lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x))

Combining (6) and (7) we obtain

∫_{0}^{1} lim inf_{n→∞} \frac{1}{n} H(g_{n}^{−1}Q_{t} ∨ ⋯ ∨ g_{1}^{−1}Q_{t}) ≤ d · lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x))

and hence for a generic t_{0} ∈ [0, 1] the partition P = Q_{t_{0}} satisfies

\underline{h}(g_{1}, g_{2}, …, P) ≤ d · lim sup_{n→∞} \frac{1}{n} log (∫_{M} ∥D(g_{n}, x)∥ dω(x)) .

Thereby proving the theorem.

4. Proof of Theorem 7.

A vector u ∈ L^{2}(X, μ) is said to be compact if its orbit G u is totally bounded. It is easy to see that if the G-action on (X, μ) has a discrete spectrum, then all vectors are compact. For the proof of the converse implication see [BR].

Suppose that the G-action on (X, μ) does not have a discrete spectrum. Then there exists a non compact vector u. As the set of compact vectors forms a closed subspace, we can assume u = 1_{E} for some E ⊂ X. Let (g_{1}, g_{2}, …) be any G-sequence, s.t. {g_{i} u}_{i=1}^{∞} are 3ϵ-separated, for some ϵ > 0.

Now we take the partition P = (E, X \ E) (recall that u = 1_{E}), and construct inductively a subsequence \{h_{j}\} of \{g_{i}\}, s.t. \underline{h}(h_{1}, h_{2}, …, P) > 0. Suppose that h_{1}, …, h_{n} are chosen, let P_{n} = ⋁_{i=1}^{n} h_{i}^{−1}P and let V_{P_{n}} be the finite dimensional space of P_{n}-measurable functions. Then, by Lemma 11, there exists g_{i}, s.t. dist(g_{i} u, V_{P_{n}}) ≥ ϵ. We set h_{n+1} = g_{i}, and continue inductively. We obtain a sequence {h_{i}} with the property that

dist(h_{n} u, V_{P_{n}}) ≥ ϵ P_{n} = ⋁_{i=1}^{n} h_{i}^{−1}P (8)

Lemma 10 implies that for any k ≥ 1:

H(h_{k}^{−1}P | ⋁_{i=1}^{k−1} h_{i}^{−1}P) ≥ δ, (9)

and therefore

lim_{n→∞} \frac{1}{n} H(⋁_{i=1}^{n} h_{i}^{−1}P) ≥ lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} H(h_{k}^{−1}P | ⋁_{i=1}^{k−1} h_{i}^{−1}P) ≥ δ (10)

As (8) - (10) hold for any subsequence of \{h_{i}\}, we conclude that \underline{h}(h_{1}, h_{2}, …, P) > 0, contradicting the assumption.

The other implication, namely discrete spectrum implies \overline{h}(g_{1}, g_{2}, …) ≥ δ > 0 for all sequences, was essentially proved in [Ku1] and can be easily deduced from Lemma 10.

References

[BR] V. Bergelson and J. Rosenblatt, Mixing actions of groups, Ill. J. Math. 32 (1988), 65–80.

[Fu] A. Furman, Cocycle growth and random entropy, preprint.

[Ka1] A. Katok, Counterexamples in ergodic theory, unpublished.

[Ku1] A. G. Kushnirenko, An upper bound of the entropy of a classical dynamical system, Soviet Math. Doklady [6] (1965), 360–362.

[Ku2] A. G. Kushnirenko, On metric invariants of entropy type, , 53–61

[Ma] G. W. Mackey, Ergodic transformation groups with a pure point spectrum, Ill. J. Math. 6 (1962), 327–335.

[vNH] J. von Neumann and P. R. Halmos, Operator methods in classical mechanics, Ann. of Math. (2) 43 (1942), 332–350.

[Zi1] R. J. Zimmer, Volume preserving actions of lattices in semisimple groups on compact manifolds, Publ. Math. I.H.E.S. 59 (1984), 5–33.

[Zi2] R. J. Zimmer, Spectrum, entropy and geometric structures for smooth actions of Kazhdan groups, Israel. J. Math. 75 (1991), 65–80.

Department of Mathematics University of Chicago 5734 University Ave. Chicago, IL 60637 e-mail: furman math.uchicago.edu