Absolutely Continuous Spectrum for Parabolic Flows/Maps

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.


Motivation
Spectral theory of dynamical systems has long been studied [13]; of particular interest, is the notion of when to expect the presence of absolutely continuous spectral measures. Since absolutely continuous spectrum implies mixing, this property can be thought of as an indicator of how chaotic, or how far from orderly, a system is. In the hyperbolic setting, systems are characterized as having a correlation decay that is exponential. As a result, techniques derived from the existence of a spectral gap as well as probabilistic tools are available for the study of spectral properties, and therefore, it is in the hyperbolic setting where the existence of absolutely continuous spectrum predominantly occurs. Interestingly, certain parabolic systems also share this property despite having at most polynomial decay of correlations. This slower decay of correlations precludes the use of the tools available in the spectral study of hyperbolic systems, and consequently, spectral theory of smooth parabolic flows and smooth perturbations of well known parabolic flows has been much less studied. This work is devoted to creating an abstract framework for the study of certain spectral properties of parabolic systems. Specifically, we attempt to answer the question: under what general conditions can we expect the existence of absolutely continuous spectral measures?

Statement of results
In Theorem 1 in we present general conditions under which we expect a skew-adjoint operator to have absolutely continuous spectrum.
The proof of this theorem is a general application of the method in [5], in which the authors show that the Fourier Transform of the spectral measures of smooth coboundaries are square integrable. The method in [5] was inspired by Marcus's proof of mixing of horocycle flows [14] which requires a specific form of tangent dynamics from which one can exploit shear of nearby trajectories. In addition, we rely on the bootstrap technique from [5] to estimate the decay of correlations of specific smooth coboundaries. Our choice of coboundaries depends upon a growth condition involving the commutator of the skew-adjoint operator and a certain auxiliary operator. We use this general, functional analytic result to derive the following results for certain parabolic dynamical systems. Theorem 2 states that time-changes of unipotent flows on homogeneous spaces of semisimple groups have absolutely continuous spectrum. In the compact case, we also show that the maximal spectral type is Lebesgue, following the method in [5]. The time-changes of the horocycle flow are special cases of this general category of flows, and spectral properties of the time-changes of horocycle flows were shown in [5], [18], and [19]. In addition, we use the general conditions to prove Theorem 4 and Corollary 2 regarding spectral results for twisted horocycle flows, combining the horocycle time-change with a circle rotation. Lastly, we rederive certain spectral results for skew products over translations and Furstenberg transformations, originally shown in [19]. Our results are slightly weaker as we prove results for functions of class C 2 while the author in [19] considers functions of class C 1 with an added Dini Condition. Remark 1. It is standard to consider spectral decomposition in the setting of self-adjoint operators. When we consider flows that are represented as strongly continuous one-parameter unitary groups, the generating operators (vector fields) are essentially skew-adjoint. Since multiplication by i gives an essentially self-adjoint operator we make no distinction from the standard setting, and thus, directly apply the theory for self-adjoint operators.

Abstract conditions 2.1 Preliminary assumptions
Suppose that a closed operator X on a Hilbert Space H, defined on a dense subspace D such that X(D) ⊂ D, generates a strongly continuous, one parameter group {e sX }.
Suppose also that e tU (D) ⊂ D is a strongly continuous, unitary group with infinitesimal generator U , and that the commutator . Theorem 1. If for β > 1 2 , H(t) and H satisfy:  [5] which was based on Marcus's shear mechanism in [14]. The authors in [20] and [17] have used a similar term to derive criteria for strong mixing.
Proof. Let f ∈ Ran(H) and letμ f (t) = R e itξ dµ f (ξ) be the Fourier Transform of the spectral measure µ f . .
For s ∈ [0, σ], we integrate by parts: e sX e tU f σ 0 e sX e tU f ds.  .

I. LetH
(s, t) = e sX e tU I −H We compute, for f = e sX (e tU (Hu)),   Using the identity e −tU X, e tU = − X, e −tU e tU we can simplify and combine terms: for some constants C and C 2 , and thus, ∂H(s, t) ∂s

II.
For g ∈ Dom(H), Finally, from I. and II., So for σ > 0, chosen such that 0 < C 1 + σC 2 < 1, for all t sufficiently large, Thus, sinceμ f (t) ∈ L 2 (R), µ f is absolutely continuous. As this holds for f ∈ Ran(H), µ f is absolutely continuous for any f ∈ H.
Remark 4. The proof for the discrete case is the same aside from the replacement of the continuous parameter t and norm · L 2 (R) by the discrete parameter n and norm · ℓ 2 (Z) . The conclusion becomes for β > 1 2 , and thus, µ f (n) ∈ ℓ 2 (Z).

Applications to flows
3.1 Time-changes of unipotent flows on homogeneous spaces of semisimple groups As a direct consequence of Theorem 1, we derive a result for a specific category of generating operators.
Let G be a semisimple Lie group and let the manifold M = Γ \ G for some lattice Γ in G such that M has finite area.
By the Jacobson-Morozov Theorem, any nilpotent element U of the semisimple Lie algebra of G is contained in a subalgebra isomorphic to sl 2 . This means that this subalgebra contains an element X, such that [U, X] = U . Let e tU be a unitary operator of the Hilbert space L 2 (M, vol). Thus, if the unipotent flow generated by U , , is ergodic, then from Lemma 5.1 in [15], it has purely absolutely continuous spectrum on Let α : M → R + , be the infinitesimal generator of τ , such that α ∈ C ∞ (M ) and Let e tUα be a unitary operator on the Hilbert space L 2 (M, vol α ). The following formulas hold on The ergodicity of φ Uα t gives us the following limit a.e., From the Dominated Convergence Theorem, with dominating function 2 G(α) ∞ , we have convergence in So if we integrate both sides of with respect to s, we obtain the following equality Thus, the preliminary assumptions for Theorem 1 are satisfied with B 1 = 1 α I and B 2 = αI. Theorem 2. a. Any smooth time-change of an ergodic flow on M generated by a nilpotent element of a semisimple Lie algebra has absolutely continuous spectrum on Any smooth time-change of a uniquely ergodic flow on M generated by a nilpotent element of a semisimple Lie algebra has absolutely continuous spectrum on L 2 0 (M, vol α ). Remark 5. The condition in part a. is equivalent to the condition employed by Kushnirenko [12] (Theorem 2) to prove mixing for the time-changes of the horocycle flow. As shown by Marcus, this condition is unnecessarily restrictive. In the compact setting, the authors in [5] prove spectral results using the implicit unique ergodicity instead of requiring such a condition. The author in [18] proves similar spectral results by imposing this Kushnirenko condition; the author later substitutes this condition by a utilization of unique ergodicity [19] in the compact case. In the noncompact setting it remains open as to whether or not spectral results can be derived without imposing a Kushnirenko-type condition.
Proof. We show that the conditions of Theorem 1 hold.
Since the above holds for f ∈ Ran(U α ), H(t) t H −1 extends to a bounded operator on Ran(U α ) = L 2 0 (M, vol α ) with uniformly bounded norm in t, Also, Since the above holds on Ran(U α ) the following is true on Ran(U α ) = L 2 0 (M, vol α ), (ii) In the following calculation we use that where Dφ Uα t denotes the differential of the diffeomorphism φ Uα t . X, and obtain the bound, where C ′′ α depends on the second derivative of α. Since is the multiplication operator given by we obtain the following bound, Thus, X, H(t) t H −1 extends to a bounded operator on Ran(U α ) with operator norm uniformly bounded in t: b. Now we assume that the flow {φ U t }, and hence {φ Uα t }, are uniquely ergodic.
If {φ Uα t } is uniquely ergodic, then the following converges uniformly, and thus, Hence, is satisfied on Ran(U α ) = L 2 0 (M, vol α ) without imposing any further conditions on Xα α . The remainder of the proof is the same as in a except that where C ′ α is finite but not necessarily equal to 1.
Theorem 3 (Maximal Spectral Type). The maximal spectral type of the uniqely ergodic flow {φ Uα t } is Lebesgue on the subspace Ran(U α ).
Proof. We follow the method in [5]. Lemma 1. [5] Suppose that the maximal spectral type of {φ Uα t } is not Lebesgue. Then there exists a smooth non-zero function ω ∈ L 2 (R, dt) such that for all functions g ∈ C ∞ (M ) the following holds: R ω(t) σ 0 e sX e tUα U α g ds dt = 0 Proof. Since the maximal spectral type is not Lebesgue, then there exists a compact set A ⊂ R such that A has positive Lebesgue measure but measure 0 with respect to the maximal spectral type. So we let ω ∈ L 2 (R) be the complex conjugate of the Fourier transform of the characteristic function χ A of the set A ⊂ R. For f, h ∈ Ran(U α ), let µ f,h denote the joint spectral measure (which we know is absolutely continuous with respect to Lebesgue since f, h ∈ Ran(U α ). Thus, In particular, when f = U α g we have Let g ∈ C ∞ (M ) such that g = 0 on M \ Im(E T ρ,σ ) and Let T ρ,σ > 0 be defined as: From unique ergodicity, lim ρ→0 + T ρ,σ = +∞.
The composition of the flow box with U α g and Xg follow from the commutation relations: From the assumptions of Lemma 1 and by integrating 1, we have The bound C σ (α) of σ 0 e sX e tUα U α gds derived for the spectral results, combined with 1 and 2, give us the following L 2 bound, Since the above bound is uniform with respect to ρ, we can conclude that the following limit holds, Combining equation 2 with the limit result in 3 implies that R ω(t) dψ(t) dt dt = 0 and thus, ω ≡ 0.

Time changes of the horocycle flow -compact and finite area
On M = Γ \ P SL(2, R), where M is either compact or of finite area, we consider the basis of the Lie algebra sl 2 (R), where U and V are the generators of the positive and negative horocycle flows, {h U t } and {h V t } respectively, and X is the generator of the geodesic flow, {φ X s }. From [1], we know that iU, iV, iX are essentially self-adjoint on C ∞ (M ), and thus, U, V, X are essentially skew-adjoint on C ∞ (M ). It follows that time-changes of the horocycle flow are special cases of Theorem 2 (when M is of finite volume, {h Uα t } is ergodic, and when M is compact, {h Uα t } is uniquely ergodic). The following Corollary was already proved in [5], [18], [19] under slightly weaker regularity assumptions; in this paper we have not attempted to optimize the regularity.

Twisted horocycle flows
Much work has been done on the spectral analysis of skew products on tori, for example, [3], [9], [10], [11]. We would like to consider a skew product for which the base dynamics are ergodic (in fact uniquely ergodic), but not an action on S 1 . For such an example, we will examine the conditions under which the spectral properties persist or do not persist after we combine the horocycle time-change with a circle rotation. Our new space iŝ M = (Γ \ P SL(2, R)) × S 1 for Γ a cocompact lattice. We define the following operators: Proof. Consider the time-change {φ Wα t } = 1 α W =Û α ×d dθ . Since {h Uα t } is mixing [14], then it is weakly mixing, and thus {φ Wα t } is ergodic [4]. This implies the ergodicity of {φ W t }. Since {φ W t } is ergodic and {h U t } is uniquely ergodic [7], then from [6] (applied to flows), {φ W t } is uniquely ergodic. We are interested in the spectrum of the flow {φ W t }, so we compute the commutator withX. Since, e sX e tW f, e sX f ds , the preliminary assumptions are satisfied with B 1 = B 2 = I. However, if we proceed with verifying the conditions of Theorem 1 for functions in the range of H, we are unable to extend pointwise bounds in L 2 (M ) to uniform bounds in the operator norm. Instead we modify our operators by introducing an operator P , defined in such a way that it not only acts as a projection operator but also preserves regularity.
Let χ ∈ C ∞ 0 (R \ {0}) such that the support of χ is a compact subset of the spectrum of H away from 0. For f, g ∈ L 2 (M ), since H is a vector field, and thus, The decay of e tH f = f • φ H t is at most polynomial in t, however, since χ ∈ C ∞ 0 (R \ {0}),χ ∈ S(R), and thus, must decay faster than any power of 1 t . In this way, we guarantee that and we take D = P (C ∞ (M )).
Now we introduce our modified operators. LetX p = PXP. Since P commutes with e tH , P commutes with H. Thus, P commutes with W and e tW . Therefore, e −tW X P , e tW = P e −tW X , e tW P = P H(t)P = H P (t).
For u ∈ C ∞ (M ), lim t→∞ H P (t) t u = P HP u = HP 2 u = H P u.
Note that now H P is a bounded, invertible operator. Let Proof. We will verify the conditions of Theorem 1 on each subspace Also, Since {φ W t } is uniquely ergodic, the following converges uniformly, and hence, Before we bound terms a-e, we show bounds for the terms X , P and X , H(t) t P .
X , P f = The first term we integrate by parts: The boundedness of the second term follows immediately, Thus, X , P f and hence, X , P op ≤ C.
Also, X , H(t) t P = X , W P + X , Now we bound the following term, SinceXα − α is a function on M , We consider the L ∞ (M ) norm and let Thus, X , H(t) t P extends to a bounded operator on Ran(H p ) with operator norm uniformly bounded in t: : (iii).
[H P (t), We have shown that the conditions of Theorem 1 are satisfied on Ran(P ). We would like to extend this to Ran(H). Recall that H P depends upon a choice of χ ∈ C ∞ 0 (R \ {0}). For f ∈ Dom(H), we can express the following in terms of integrals involving the spectral projector as Let χ be such that and supp(χ) vanishes outside of I ǫ,K . Since on I ǫ,K , we consider H P f − Hf on R \ I ǫ,K , i.e., =0.
For |x| ≥ K, Thus, inf Consequently, for every f ∈ Ran(H), µ f is absolutely continuous.
The characteristics of Ran(H) are linked to the properties of the cocycle For π a projection from R to S 1 , letâ = π(a).
, for k ∈ Z + and g : M → S 1 .
From [3] we have the following cases. If k 0 = ∞ then {φ W t } has purely absolutely continuous spectrum on L 2 0 (M ); this is the case when {φ H t } is ergodic, and thus, Ran(H) = L 2 0 (M ).
When, k 0 < ∞, we have a nontrivial pure point component of the spectrum. To show this, we consider the subspaces given by for f ∈ L 2 (M ) and n ∈ Z.
The operator We compute H t • V t : Hence, on E nk0 , H t is unitarily equivalent to S t . Since e ink0θ is an invariant function for S t , H t has an eigenfunction in E nk0 for every n. Thus, the spectrum on E nk0 has an absolutely continuous component as well as an infinite dimensional pure point component. This leads to the following Corollary.

Applications to maps
The author in [19] uses the Mourre Estimate [2], [16] to prove the following spectral results. Here we rederive similar results by showing that the conditions of Theorem 1 are satisfied; our results are slightly weaker as we prove results for functions of class C 2 while the author in [19] considers functions of class C 1 with an added Dini Condition. We use the notation and description from [19].

Skew products over translations
Let X be a compact metric abelian Lie group with normalized Haar measure µ. Let {F t } be a uniquely ergodic [6] translation flow (we assume that F 1 is ergodic), The associated operators {V t } are given by Let G be a compact metric abelian group. Let φ : X → G such that φ can be written as φ = ξη where ξ is a group homomorphism and η satisfies forη χ ∈ Dom(P ) a real-valued function determined by χ and η. The skew product, T : X × G → X × G, is defined by with corresponding unitary operator LetĜ be the character group of G. The decomposition L 2 (X × G) = χ∈Ĝ L χ , L χ = {ϕ ⊗ χ : ϕ ∈ L 2 (X)} and the restriction of W to the subspaces L χ allow us to study the spectrum of convenient, unitarily equivalent operators to W | Lχ , namely, (Here U χ takes the place of e U as given in the conditions.) We will choose to take the commutator with P ; it follows from [1] that P is essentially self-adjoint on D = C ∞ (X).
Thus, Thus, condition (iii) is immediately satisfied. Since conditions (i), (ii), and (iii) of Theorem 1 are satisfied on each L χ , we have shown that the operator U χ has purely absolutely continuous spectrum on L χ . Thus, W has purely absolutely continuous spectrum when restricted to the subspace χ∈Ĝ,χ•ξ ≡1 L χ .
In addition, from the purity law in [8] extended to translations, the maximal spectral type is either purely Lebesgue, purely singularly continuous, or purely discrete with respect to µ (the Haar measure). Since we know that the spectrum is absolutely continuous from above, we have proved the following theorem.
Similar results with less restrictive assumptions on η have been derived in [11] and [19].
Thus, condition (iii) is immediately satisfied. Since conditions (i), (ii), and (iii) of Theorem 1 are satisfied, the operator U j,k has purely absolutely continuous spectrum on each H j,k . Thus, W d has purely absolutely continuous spectrum on the orthocomplement of H 1 . Applying the the purity law in [8], we derive, with slightly stronger regularity assumptions, a similar result to the ones found in [11] and [19]. Theorem 6. W d has countable Lebesgue spectrum on the orthocomplement of H 1 .