FURSTENBERG S THEOREM ON PRODUCTS OF I.I.D. 2 ˆ 2 MATRICES JAIRO BOCHI Abstract. This is a revised version of some notes written ą 0 years ago. I thank Anthony Quas for pointing to a gap in the previous proof of Lemma 5 and for providing a correct proof. These notes follow [BL]. We deal with Lyapunov exponents of products of random i.i.d. 2 ˆ 2 matrices of determinant. Let SL p2, Rq denote the group of such matrices. Let µ be a probability measure in SL p2, Rq which satisfies the integrability condition log }M} dµpmq ă 8. SL p2,rq If Y, Y 2,... are random independent matrices with distribution µ, then the limit γ lim nñ8 n log }Y n Y } (the upper Lyapunov exponent) exists a.s. and is constant, by the subadditive ergodic theorem. We have γ ě 0. The Furstenberg theorem says that γ ą 0 for most choices of µ. Let us see some examples where γ 0: () If µ is supported in the orthogonal group Op2q then γ 0. (2) If µ is supported in the abelian subgroup * "ˆt 0 0 t ; t P Rzt0u then γ ş log }M} dµpmq, which may be zero. (3) Assume that only two matrices occur: ˆ ˆ2 0 0 and R 0 {2 π{2. 0 Then it is a simple exercise to show that γ 0. Furstenberg s theorem says that the list above essentially covers all possibilities where the exponent vanishes: Date: May 9, 206. Note that }M} }M } ě if M P SL p2, Rq.
2 FURSTENBERG S THEOREM Theorem. Let µ be as above, and let G µ be the smallest closed subgroup which contains the support of µ. Assume that: (i) G µ is non compact. (ii) There is no finite set H L Ă P such that MpLq L for all M P G µ. Then γ ą 0. Remark. Condition (i) is equivalent to: (i ) There is no C P GLp2, Rq such that CMC is an orthogonal matrix, for all M P G µ. Remark. Under the assumption (i), condition (ii) is equivalent to: (ii ) There is no set L Ă P with #L or 2 and such that MpLq L for all M P G µ. (This follows from the fact that if M P SL p2, Rq fixes three different directions then M I.) Non-atomic measures in P Let MpP q be the space of probability Borel measures in P. A measure ν P MpP q is called non-atomic if νptxuq 0 for all x P P. We collect some simple facts for later use. If A P GLp2, Rq then we also denote by A the induced map A: P Ñ P. If A in not invertible but A 0 then there is only one direction x P P for which Ax is not defined. In this case, it makes sense to consider the push-forward Aν P MpP q, if ν P MpP q is non-atomic. Lemma. If ν P MpP q is non-atomic and A n is a sequence of non-zero matrices converging to A 0, then A n ν Ñ Aν (weakly). The proof is easy. Lemma 2. If ν P MpP q is non-atomic then is a compact subgroup of SL p2, Rq. H ν tm P SL p2, Rq; Mν νu Proof. Assume that there exists a sequence M n in H ν with }M n } Ñ 8. Up to taking a subsequence, we may assume that the sequence (of norm matrices) }M n } M n converges to a matrix C. Since C 0, Lemma gives Cν ν. On the other hand, det C lim }M n } 2 0. Thus C has rank one and ν Cν must be a Dirac measure, contradiction.
FURSTENBERG S THEOREM 3 µ-stationary measures in P If ν P MpP q, let the convolution µ ν P MpP q is the push-forward of the measure µˆν by the natural map ev: SL p2, RqˆP Ñ P. If µ ν ν then ν is called µ-stationary. By a Krylov Bogolioubov argument, µ-stationary measures always exist. Lemma 3. If µ satisfies the assumptions of Furstenberg s theorem then every µ-stationary ν P MpP q is non-atomic. Proof. Assume that β max νptxuq ą 0. xpp Let L tx P P ; νptxuq βu. If x 0 P L then β νptx 0 uq pµ νqptx 0 uq χ tx0 upmxq dµpmq dνpxq νptm px 0 quq dµpmq. But νptm px 0 quq ď β for all M, so νptm px 0 quq ď β for µ-a.e. M. We have proved that M plq Ă L for µ-a.e. M. This contradicts assumption (ii). From now on we assume that µ satisfies the assumptions of Furstenberg s theorem, and that ν is a (non-atomic) µ-stationary measure in P. ν and γ The shift σ : SL p2, Rq N Ðâ in the space of sequences ω py, Y 2,...q has the ergodic invariant measure µ N. Consider the skew-product map T : SL p2, Rq NˆP Ðâ, T pω, xq pσpωq, Y pωqxq. Consider f : SL p2, Rq N ˆ P Ñ R given by fpω, xq log }Y pωqx}. (The notation is obvious). Then nÿ fpt j pω, xqq n n log }Y npωq Y pωqx}. by Oseledets theorem, for a.e. ω and for all x P P ztepxqu, 2 the quantity on the right hand side tends to γ as n Ñ 8. In particular, this convergence holds for µ N ˆ ν-a.e. pω, xq. We conclude that () γ f dµ N dν log }Mx} dµpmq dνpxq. 2 Epxq is the direction associated to the exponent γ, if γ ą 0.
4 FURSTENBERG S THEOREM Convergence of push-forward measures Let S n pωq Y pωq Y n pωq. Lemma 4. For µ N -a.e. ω, there exists ν ω P MpP q such that S n pωqν Ñ ν ω. Proof. Fix f P CpP q. Associate to f the function F : SL p2, Rq Ñ R given by F pmq fpmxq dνpxq. Let F n be the σ-algebra of SL p2, Rq N formed by the cylinders of length n; then S n p q is F n -measurable. Also EpF ps n` q F n q F ps n Mq dµpmq fps n Mxq dµpmq dνpxq fps n yq dνpyq F ps n q (since µ ν µ). This shows that the sequence of functions ω ÞÑ F ps n pωqq is a bounded martingale. Therefore the limit Γfpωq lim nñ8 F ps n pωqq exists for a.e. ω. Now let f k ; k P N be a countable dense subset of CpP q. Take ω in the full-measure set where Γf k pωq exists for all k. Let ν ω be a (weak) limit point of the sequence of measures S n pωqν. Then f k dν ω lim nñ8 f k dps n νq lim nñ8 f k S n dν Γf k pωq. Since the limit is the same for all subsequences, we have in fact that S n pωqν Ñ ν ω. Let s explore the construction of the measures to obtain more information about them: Lemma 5. The measures ν ω from Lemma 4 satisfy S n pωqmν Ñ ν ω as n Ñ 8 for µ-a.e. M. Proof. We show that for any fixed f k from the sequence above that for µ-a.e. M, that ş f k ps n pωqmxq dνpxq Ñ ş f k dν ω pxq for µ N almost every ω. Let F k : SL p2, Rq Ñ R be the function introduced above corresponding to f k. Given this, by taking the intersection over countably many sets ş of full µ-measure, we obtain a set of M s of full measure on which fpsn pωqmxq dνpxq Ñ ş f dν ω pxq for µ N -a.e. ω for all f P CpP q, that is, weak convergence of S n pωqmν to ν ω.
We consider the expression «ÿ 8 ˆ I E f k ps n pωqmxq dνpxq n FURSTENBERG S THEOREM 5 2ff f k ps n pωqxq dνpxq dµpm q, where E denotes integration in the ω variable with respect to µ N. We establish below that I ă 8. From this is follows that the quantity in the brackets is finite for µ N -a.e. ω and µ-a.e. M. It follows that for µ N -a.e. ω and µ-a.e. M, ş f k ps n pωqmxq dνpxq ş f k ps n pωqxq dνpxq Ñ 0. However for µ N -a.e. ω, we have ş f k ps n pωqxq dνpxq Ñ ş f k pxq dν ω pxq. Combining these, we obtain the desired result. To prove that I ă 8, we note that 8ÿ ˆ 2 I E f k ps n pωqmxq dνpxq f k ps n pωqxq dνpxq dµpm q. n Now define I n by ˆ I n E f k ps n pωqmxq dνpxq E F k ps n pωqmq F k ps n pωq 2 dµpmq 2 f k ps n pωqxq dνpxq dµpmq E F 2 k ps n pωqmq 2 2F k ps n pωqmqf k ps n pωqq ` F k ps n pωqq dµpm q. Notice that the distribution of conditional distribution of S n` pωq given S n pωq is the same as that of S n pωqm. Hence I n E F 2 k ps n` pωqq 2 2F k ps n` pωqqf k ps n pωqq ` pf k ps n pωqq. Notice E F k ps n` ωqf k ps n pωq E E ˇ E F k ps n` ωqf k ps n pωqˇf n F k ps n pωqqepf k ps n` pωqq F n q EpF k ps n pωqq 2, where we used the tower law for conditional expectations for the second equality. Hence I n EF k ps n` pωqq 2 EF k ps n pωqq 2. Now I lim Nÿ NÑ8 n I n lim NÑ8 EF kps N` pωqq 2 EF k ps pωqq 2 ď }f k } 2. The limit measures are Dirac Lemma 6. For µ N -a.e. ω, there exists Zpωq P P such that ν ω δ Zpωq. Proof. Fix a µ N -generic ω. By Lemma 5 we have, for µ-a.e. M, lim S n ν lim S n Mν.
6 FURSTENBERG S THEOREM Let B be a limit point of the sequence of norm matrices }S n } S n. Since }B}, we can apply Lemma : Bν BMν. If B were invertible, this would imply ν Mν. That is, a.e. M belongs to the compact group H ν (see Lemma 2) and therefore G ν Ă H ν, contradicting hypothesis (i). So B is non-invertible. Since Bν ν ω, we conclude that ν ω is Dirac. Convergence to Dirac implies norm growth Lemma 7. Let m P MpP q be non-atomic and let pa n q be a sequence in SL p2, Rq such that A n m Ñ δ z, where z P P. Then Moreover, for all v P R 2, }A n } Ñ 8. }A npvq} }A n } Ñ xv, zy. Proof. We may assume that the sequence A n {}A n } converges to some B. Since }B}, we can apply Lemma to conclude that Bm δ z. If B were invertible then we would have that m δ B z would be atomic. Therefore det B 0 and }A n } 2 ˇ ˇdet A n }A n } ˇ Ñ det B 0. So }A n } Ñ 8. Notice that the range of B must be the z direction. Let v n, u n be unit vectors such that A n v n }A n }u n. Then u n A npv n q }A n }. Since A n {}A n } Ñ B and }B}, we must have u n Ñ z (up to changing signs). Moreover, u n is the direction which is most expanded by A n. The assertion follows. (For a more elegant proof, see [BL, p. 25].) Convergence to 8 cannot be slower than exponential We shall use the following abstract lemma from ergodic theory: Lemma 8. Let T : px, mq Ðâ be a measure preserving transformation of a probability space px, mq. If f P L pmq is such that then ş f dµ ą 0. n ÿ fpt j xq `8 for m-almost every x,
FURSTENBERG S THEOREM 7 Proof. 3 For any function g, let g denote the limit of Birkhoff averages of g. Then f ě 0. Assume, for a contradiction, that ş f 0. Then f 0 a.e. Let s n ř n f T j. For ε ą 0, let A ε tx P X; s n pxq ě ε @n ě u and B ε ď kě0 T k pa ε q. Fix ε ą 0 and let x P B ε. Let k kpxq ě 0 be the least integer such that T k x P A ε. We compare the Birkhoff sums of f and χ Aε : n ÿ fpt j xq ě k ÿ n ÿ fpt j xq ` Dividing by n and making n Ñ 8 we get j k 0 fpxq ě εąχ Aε pxq εχ Aε pt j xq @n ě. Therefore µpa ε q Ąχ Aε Ąχ Aε 0. B ε Thus µpb ε q 0 for every ε ą 0 as well. On the other hand, if s n pxq Ñ 8 then x P Ť εą0 B ε. We have obtained a contradiction. End of the proof of the theorem. Replace everywhere Y i by Yi. Note that µ also satisfies the hypothesis of the theorem if µ does. 4 Let T and f be as in page 3. By Lemmas 6 and 7 we have nÿ fpt j pω, xqq log }S npωqx} Ñ 8 for a.e. ω and all x P P ztzpωq K u. In particular, convergence holds µ N ˆ ν- a.e. By Lemma 8, this implies ş f ą 0. Then, by (), γ ą 0. References [BL] P. Bougerol and J. Lacroix. Products of random matrices with applications to Schrödinger operators. Birkhäuser, 985. [F] H. Furstenberg. Non-commuting random products. Trans. AMS, 08: 377 428, 963. 3 This proof is a bit simpler than that in [BL]. 4 Because Apvq w ñ A pw K q v K.