OON CONVERGENCE OF RANDOM WALKS ON MODULISPACE
ROLAND PROHASKA
Abstract.
The purpose of this note is to establish convergence of random walkson the moduli space of Abelian differentials on compact Riemann surfaces in twodifferent modes: convergence of the n -step distributions towards the normalizedMasur–Veech measure from almost every starting point, and almost sure path-wise equidistribution towards the affine invariant measure on the SL ( R )-orbitclosure of an arbitrary starting point. These are analogues to previous resultsfor random walks on homogeneous spaces. Introduction
Consider the moduli space of unit-area Abelian differentials on compact Riemannsurfaces, that is the space of pairs (
M, ω ), where M is a compact Riemann surfaceand ω a holomorphic 1-form on M with vol( M, ω ) = i2 R M ω ∧ ¯ ω = 1, up to bi-holomorphic equivalence. The form ω determines a flat metric on M with conicalsingularities at its zeroes. Hence, a pair ( M, ω ) can alternatively be seen as a trans-lation surface . This viewpoint can be used to define a natural SL ( R )-action onthe moduli space. The moduli space is stratified by specification of combinatorialdata: the genus g of the surface M and a partition α = ( α , . . . , α n ) of 2 g − ω . Strata are not always connected butconsist of at most three connected components, which have been classified [9]. TheSL ( R )-action preserves strata and their connected components. We refer to thesurvey [13] for further background. In the following, we restrict our attention toa connected component of a stratum, which we shall denote by H throughout thearticle.Inside H there are natural lower dimensional structures, called affine invari-ant submanifolds , which are immersed submanifolds that locally look like complexsubspaces in period coordinates; see [6, Definition 1.2]. Every affine invariant sub-manifold M carries a unique ergodic SL ( R )-invariant probability measure ν M . Aparticular case is H itself together with the normalized Masur–Veech measure. Thefollowing result will serve as our motivating example. Theorem 1.1 (Eskin–Mirzakhani–Mohammadi [6]) . Let µ be an absolutely contin-uous compactly supported SO ( R ) -bi-invariant probability measure on SL ( R ) and x ∈ H . Then the orbit closure SL ( R ) x is an affine invariant submanifold M , andwe have the weak* convergence n n − X k =0 µ ∗ k ∗ δ x −→ ν M (1.1) as n → ∞ . Date : February 12, 2021.2010
Mathematics Subject Classification.
Primary 60B15; Secondary 32G15, 60G50, 22F10.
Key words and phrases.
Random walk, moduli space, spectral gap, equidistribution. a r X i v : . [ m a t h . D S ] F e b ROLAND PROHASKA
Here µ ∗ k denotes the k -fold convolution power of µ , and weak* convergence ofmeasures means convergence when the measures in question are applied to continu-ous test functions with compact support. Spelled out explicitly, the weak* conver-gence in the conclusion of the theorem above thus means that for every compactlysupported continuous function f ∈ C c ( H ) it holds thatlim n →∞ n n − X k =0 Z f ( g k · · · g x ) d µ ⊗ k ( g , . . . , g k ) = Z f d ν M . This result should be interpreted as a statement about Cesàro convergence in lawof the random walk on H given by µ . Indeed, the convolution µ ∗ n ∗ δ x is thedistribution of the location after n steps of the random walk started at x . Thepurpose of this short article is to establish two further modes of convergence forsuch random walks:(i) Generic points for non-averaged convergence:
In §2, we prove that thestronger, non-averaged, weak* convergence µ ∗ n ∗ δ x −→ ν M as n → ∞ holds for ν M -almost every starting point x . As in the homoge-neous setting (see [11, §3]), the key ingredient is the existence of a spectralgap in L ( ν M ) of the convolution operator π ( µ ) : f (cid:18) x Z f ( gx ) d µ ( g ) (cid:19) acting on measurable functions, which is a consequence of the work ofAvila–Gouëzel [2].(ii) Pathwise equidistribution:
In §3, we improve the convergence in law in(1.1) to almost sure pathwise convergence, meaning that for µ ⊗ N -almostevery sequence ( g i ) i of elements of SL ( R ) we have1 n n − X k =0 δ g k ··· g x −→ ν M as n → ∞ in the weak* topology. The argument uses techniques developedby Benoist–Quint for the homogeneous case [3] and the Lyapunov functionsconstructed in [6, Proposition 2.13].We point out that in both cases we can deal with measures more general than theones considered in Theorem 1.1. Acknowledgments.
The author would like to thank Jayadev Athreya for valuablediscussions, his encouragement to write this article, and providing numerous usefulreferences. 2.
Generic Points
Let G be a locally compact σ -compact metrizable group and X a locally compact σ -compact metrizable space on which G acts continuously. Then for any probabilitymeasure µ on G , one can define the convolution operator π ( µ ) by π ( µ ) f ( x ) = Z f ( gx ) d µ ( g )for bounded measurable functions f on X and x ∈ X . If m X is a G -invariantprobability measure on X , this gives a bounded linear operator π ( µ ) : L ∞ ( m X ) → L ∞ ( m X ) which extends to a continuous contraction on each L p -space (see [4, Corol-lary 2.2]). ANDOM WALKS ON MODULI SPACE 3
We will be interested in the existence of an L -spectral gap of this convolutionoperator in the case where G = SL ( R ), X = M is an affine invariant submanifoldof H endowed with the ergodic SL ( R )-invariant probability measure ν M , and µ isa probability measure on SL ( R ). Definition 2.1.
We say that µ has an L -spectral gap on X if the associatedconvolution operator π ( µ ) restricted to the space L ( X, m X ) of square-integrablefunctions with mean 0 has spectral radius strictly less than 1.We note that by the spectral radius formula, µ having an L -spectral gap on X can be reformulated as the requirement thatlim n →∞ n q k π ( µ ) | nL k op < . Proposition 2.2.
Suppose that the probability measure µ on SL ( R ) is not sup-ported on a closed amenable subgroup and let M be an affine invariant submanifoldof H . Then µ has a L -spectral gap on M .Proof. It is proved in [2] that the SL ( R )-action on L ( ν M ) does not admit almostinvariant vectors. Thus, the result follows from [12, Theorem C]. (cid:3) We are now ready to state and prove the following quantitative result on genericpoints for random walk convergence.
Theorem 2.3.
Let H be a connected component of a stratum of the moduli spaceof unit-area Abelian differentials on compact Riemann surfaces and M an affineinvariant submanifold carrying the ergodic SL ( R ) -invariant measure ν M . Let µ be a probability measure on SL ( R ) that is not supported on a closed amenablesubgroup. Then for ν M -almost every x ∈ M we have µ ∗ n ∗ δ x −→ ν M (2.1) as n → ∞ in the weak* topology. This convergence is exponentially fast in thesense that for every fixed f ∈ L ( ν M ) we have lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z f d( µ ∗ n ∗ δ x ) − Z f d ν M (cid:12)(cid:12)(cid:12)(cid:12) /n ≤ ρ (cid:0) π ( µ ) | L (cid:1) / (2.2) for ν M -a.e. x ∈ M , where ρ (cid:0) π ( µ ) | L (cid:1) denotes the spectral radius of π ( µ ) restrictedto L ( ν M ) . More precisely, given ρ (cid:0) π ( µ ) | L (cid:1) < α < , choose N ∈ N such that k π ( µ ) | nL k op ≤ α n for all n ≥ N . Then if we denote f = f − R f d ν M and B α,n,f = (cid:26) x ∈ M (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) π ( µ ) n f ( x ) − Z f d ν M (cid:12)(cid:12)(cid:12) ≥ α n / k f k L for some n ≥ n (cid:27) , we have the bound ν M ( B α,n,f ) ≤ α n − α (2.3) for every n ≥ N . To make sense of this statement, recall that ρ (cid:0) π ( µ ) | L (cid:1) is guaranteed to be strictlyless than 1 by Proposition 2.2. Examples of measures to which the theorem appliesare Zariski dense measures (i.e. measures whose support generates a Zariski densesubgroup of SL ( R )) and also the measures appearing in Theorem 1.1. Proof.
In view of separability of C c ( H ), the weak* convergence (2.1) will follow ifwe can prove that for a fixed function f ∈ C c ( H ) we have π ( µ ∗ n ) f = π ( µ ) n f −→ Z f d ν M ROLAND PROHASKA ν M -a.e. as n → ∞ . Since ρ (cid:0) π ( µ ) | L (cid:1) < α approach ρ (cid:0) π ( µ ) | L (cid:1) .Thus, it suffices to establish (2.3). To this end, observe first that, for every n ≥ N , (cid:13)(cid:13)(cid:13)(cid:13) π ( µ ) n f − Z f d ν M (cid:13)(cid:13)(cid:13)(cid:13) L = k π ( µ ) n f k L ≤ k π ( µ ) | nL k op k f k L ≤ α n k f k L . By Chebyshev’s inequality, it follows that for n ≥ N we have ν M (cid:18)(cid:26) x ∈ X (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) π ( µ ) n f ( x ) − Z f d ν M (cid:12)(cid:12)(cid:12)(cid:12) ≥ α n/ k f k L (cid:27)(cid:19) ≤ k π ( µ ) n f − R f d ν M k L α n k f k L ≤ α n . Summing over n ≥ n gives the bound (2.3). (cid:3) Pathwise Equidistribution
In this section, we aim to prove the following theorem. For the statement, recallthat a measure µ on a linear group G is said to have finite exponential moments iffor δ > g
7→ k g k δ is µ -integrable, where k·k denotesany matrix norm. Theorem 3.1.
Let H be a connected component of a stratum of the moduli spaceof unit-area Abelian differentials on compact Riemann surfaces, let x ∈ H and M = SL ( R ) x be the minimal affine invariant submanifold containing x endowedwith its ergodic SL ( R ) -invariant measure ν M . Moreover, let µ be an SO ( R ) -right-invariant probability measure on SL ( R ) with finite exponential moments satisfying µ (SO ( R )) = 0 . Then for µ ⊗ N -a.e. sequence ( g i ) i we have n n − X k =0 δ g k ··· g x −→ ν M as n → ∞ in the weak* topology. The first step towards the proof of results such as Theorem 1.1 or Theorem 3.1always is the classification of (ergodic) µ -stationary measures.Recall that given a continuous group action of G on X (as at the beginning of§2), a probability measure ν on X is called µ -stationary if µ ∗ ν = ν , which meansin more detail that Z Z f ( gx ) d µ ( g ) d ν ( x ) = Z f d ν for every bounded measurable function f on X .In [5], Eskin–Mirzakhani prove that µ -stationary measures are necessarily affinewhen µ is absolutely continuous and SO ( R )-bi-invariant. In fact, what they proveis that ergodic P -invariant measures are affine, where P ⊂ SL ( R ) denotes theupper triangular subgroup, and then use that µ -stationary measures are in corre-spondence with P -invariant measures by classical results of Furstenberg [7, 8] (seealso [10, Theorem 1.4] for a concise restatement). These results of Furstenbergapply whenever µ is admissible , meaning that supp( µ ) generates SL ( R ) as a semi-group and some convolution power µ ∗ k is absolutely continuous with respect toHaar measure on SL ( R ). We can therefore record the following. Theorem 3.2 ([5]) . Suppose that µ is admissible in the sense above. Then anyergodic µ -stationary probability measure on H is affine. ANDOM WALKS ON MODULI SPACE 5
Let us quickly convince ourselves that this result applies to the measures inTheorem 3.1.
Corollary 3.3. If µ is SO ( R ) -right-invariant and satisfies µ (SO ( R )) = 0 , thenany ergodic µ -stationary probability measure on H is affine.Proof. We consider the
KAK -decomposition of SL ( R ), where K = SO ( R ) and A = (cid:26) a t := (cid:18) e t
00 e − t (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t ∈ R (cid:27) . Then, in view of K -right-invariance of µ , we can decompose µ as µ = Z µ K,t ∗ δ a t ∗ m K d η ( t ) , (3.1)where η is a probability measure on R , µ K,t is a probability measure on K forevery t ∈ R , δ a t denotes the Dirac mass at a t ∈ A , and m K is the Haar probabilitymeasure on K . The assumption that µ ( K ) = 0 then translates to the statementthat η ( { } ) = 0. It is not difficult to see, e.g. by hyperbolic geometry considerations,that m K ∗ a s ∗ m K ∗ a t ∗ m K is absolutely continuous with respect to Haar measureon SL ( R ) whenever s, t = 0. Calculating the third convolution power using therepresentation (3.1) of µ above, it follows that µ ∗ is absolutely continuous as well.Also by hyperbolic geometry, supp( µ ) generates SL ( R ) as a semigroup. Hence, µ is admissible and Theorem 3.2 applies. (cid:3) Even though there is more to be done, let us already now give the proof of themain theorem of this section, as it will motivate the remaining work.
Proof of Theorem . By the Breiman law of large numbers (see [3, Corollary 3.3]),for µ ⊗ N -a.e. sequence ( g i ) i every weak* limit ν of the sequence ( n P n − k =0 δ g k ··· g x ) n ofempirical measures is µ -stationary. By Corollary 3.3, the measures featuring withpositive weight in the ergodic decomposition of ν can only be affine measures ν N with N ⊂ M and the point mass δ ∞ at infinity, the latter corresponding to potentialescape of mass. We will show below (Corollary 3.6 and Proposition 3.8) that δ ∞ and any given affine measure ν N with N (cid:40) M do not appear in the decompositionwith positive weight for µ ⊗ N -a.e. ( g i ) i . Knowing that only countably many suchaffine measures exist (see [6, Proposition 2.16]), the theorem follows by taking acountable intersection of full measure sets. (cid:3) We see that it remains to rule out the occurrence of unwanted limit measures.The key tool to achieve this is the following concept.
Definition 3.4.
Consider a measurable group action of G on a standard Borelspace X . A measurable function V : X → [0 , ∞ ] is called a Lyapunov function for the random walk on X induced by a probability measure µ on G if there existconstants α ∈ (0 , β ≥ π ( µ ) V ≤ αV + β , where π ( µ ) is the associatedconvolution operator introduced in §2.Intuitively speaking, the contraction inequality means that after a step of therandom walk, the value of the Lyapunov function V on average gets smaller by aconstant factor, at least outside some compact set where the value of V lies belowsome threshold depending on the additive constant β . The dynamics are thereforedirected towards the part of the space where V takes small values. The followingquantification of this phenomenon is due to Benoist–Quint. ROLAND PROHASKA
Proposition 3.5 ([3, Proposition 3.9]) . Suppose the random walk on X induced by µ admits a Lyapunov function V : X → [0 , ∞ ] . Then there exists a constant C > such that for any x ∈ X with V ( x ) < ∞ , for µ ⊗ N -a.e. ( g i ) i we have for any M > n →∞ n |{ ≤ k < n | V ( g k · · · g x ) > M }| ≤ CM .
Corollary 3.6.
Let G be a locally compact σ -compact metrizable group and X alocally compact σ -compact metrizable space endowed with a continuous G -action.Let µ be a probability measure on G and suppose that the induced random walk on X admits a Lyapunov function V : X → [0 , ∞ ] with the additional property thatfor every M > the sublevel set X M = V − ([0 , M ]) is relatively compact and itsclosure X M is contained in X \ V − ( {∞} ) . Then for µ ⊗ N -a.e. ( g i ) i , any weak*limit ν of ( n P n − k =0 δ g k ··· g x ) n satisfies ν ( X \ V − ( {∞} )) = 1 .Proof. Fix a sequence ( g i ) i such that the conclusion of Proposition 3.5 holds. Sincethe measure ν is regular, the assumptions imply that for every ε > M > f ε,M on X bounded by 1 which takes the value 1 on X M such that ν ( X M ) ≥ Z f ε,M d ν − ε. Applying weak* convergence to this function, it follows that ν ( X \ V − ( {∞} )) ≥ ν ( X M ) ≥ Z f ε,M d ν − ε ≥ − C/M − ε. Letting M → ∞ and ε → (cid:3) In view of the above, it remains to find Lyapunov functions V on H taking thevalue ∞ precisely on a given affine invariant submanifold N and satisfying theproperness conditions in Corollary 3.6. The case N = ∅ (responsible for ruling outescape of mass, i.e. the occurrence of δ ∞ as part of the limit measure) was dealtwith by Athreya [1]; most of the work necessary for the general case was carriedout in [6]. Proposition 3.7 ([6, Proposition 2.13]) . For t > , define µ t = ( a t ) ∗ m K . Let N ⊂H be an affine invariant submanifold ( N = ∅ is allowed ) . Then there exists β ≥ and an SO ( R ) -invariant function f N : H → [1 , ∞ ] with the following properties: (i) f − N ( {∞} ) = N and for every M > the closure of the sublevel set f − N ([0 , M ]) is compact and contained in H \ N , (ii) for every < α < there exists t such that for t ≥ t it holds that π ( µ t ) f N ≤ αf N + β, and (iii) for some σ > and all g in a neighborhood of the identity in SL ( R ) wehave σ − f N ( x ) ≤ f N ( gx ) ≤ σf N ( x ) for all x ∈ H . Since any g ∈ SL ( R ) is a product of at most O (log k g k ) + 1 elements of a givenneighborhood of the identity, iterating (iii) above we more generally obtain:(iii’) there exist constants σ > , κ > g ∈ SL ( R ) and x ∈ H σ − k g k − κ f N ( x ) ≤ f N ( gx ) ≤ σ k g k κ f N ( x ) . ANDOM WALKS ON MODULI SPACE 7
The final step is to use the functions provided by the above proposition to con-struct the required Lyapunov functions for the measures from the statement ofTheorem 3.1.
Proposition 3.8.
Let µ be a probability measure on SL ( R ) with finite exponentialmoments that is SO ( R ) -right-invariant and not equal to the Haar measure on SO ( R ) . Let N ⊂ H be an affine invariant submanifold. Then there exists aLyapunov function V N for µ with V − N ( {∞} ) = N that satisfies the conditions ofCorollary . The proof is an extension of the argument for [6, Lemma 3.2].
Proof.
We will first show that there exists m ∈ N such that the function f N providedby Proposition 3.7 is a Lyapunov function for the m -step random walk, i.e. with π ( µ ) m f N ≤ αf N + β for some α < , β ≥ K := SO ( R )-bi-invariant measure µ . In this case, wecan write µ ∗ m = Z R + m K ∗ δ a t ∗ m K d η ( m ) ( t )for some probability measure η ( m ) on R + . Since the random walk on SL ( R ) givenby µ has a positive top Lyapunov exponent by Furstenberg’s theorem [7], for anyfixed t > η ( m ) ([0 , t ]) → m → ∞ . From property (ii) in Proposi-tion 3.7 we know that given α ∈ (0 ,
1) there exists t such that π ( µ t ) f N ≤ α f N + β for all t ≥ t , where µ t is as defined in that proposition. Iterating property (iii) ofthe function f N , there exists some constant R > f N ( a t kx ) ≤ Rf N ( x )for all k ∈ K , 0 ≤ t ≤ t and x ∈ H . Using that f N is K -invariant we thus find π ( µ ) m f N ( x ) = Z ∞ π ( µ t ) f N ( x ) d η ( m ) ( t )= Z t π ( µ t ) f N ( x ) d η ( m ) ( t ) + Z ∞ t π ( µ t ) f N ( x ) d η ( m ) ( t ) ≤ Rη ( m ) ([0 , t ]) f N ( x ) + α f N ( x ) + β for all x ∈ H . Since as noted before, the term Rη ( m ) ([0 , t ]) tends to 0 as m → ∞ ,the right-hand side above is bounded by αf N ( x ) + β for sufficiently large m , whichis what we needed.The argument for merely K -right-invariant µ can be reduced to the case above.Indeed, if we set ˜ µ = m K ∗ µ , then what we have already established implies π (˜ µ ) m f N ≤ αf N + β for all large m , and using again K -invariance of f N we see π ( µ ) m f N ( x ) = Z f N ( gx ) d µ ∗ m ( g ) = Z f N ( gx ) d( m K ∗ µ ∗ m )( g ) = π (˜ µ ) m f N ( x )for all x ∈ H , since m K ∗ µ ∗ m = ˜ µ ∗ m .Finally, let κ be the constant from property (iii’) after Proposition 3.7 and, usingthat µ has finite exponential moments, choose δ ∈ (0 ,
1) such that R k g k κδ d µ ( g ) < ∞ . We define V N = m − X k =0 α δ ( m − − k ) m π ( µ ) k f δ N . Then, using the contraction property of f N established above and Jensen’s inequal-ity, V N can be seen to satisfy a contraction property with respect to the measure µ for the constants α δ/m and β δ . We also have V − N ([0 , M ]) ⊂ f − N ([0 , α (1 − m ) /m M /δ ]),so that the closure of the sublevel set V − N ([0 , M ]) is compact and contained in H\N
ROLAND PROHASKA due to the corresponding property of f N . It remains to argue that V − N ( {∞} ) = N .The inclusion “ ⊃ ” is clear. To see that also the reverse inclusion holds, we useproperty (iii’) of the function f N to obtain that π ( µ ) k f δ N ( x ) ≤ σ δ Z k g k κδ d µ ∗ k ( g ) f δ N ( x ) ≤ σ δ (cid:18)Z k g k κδ d µ ( g ) (cid:19) k f δ N ( x ) < ∞ for 0 ≤ k < m and any x ∈ H \ N , by choice of δ and since f − N ( {∞} ) = N . Hence,for x ∈ H \ N we also have V N ( x ) < ∞ . This finishes the proof. (cid:3) References [1] Jayadev S. Athreya. Quantitative recurrence and large deviations for Teichmuller geodesicflow.
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Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
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