ar X iv :1 81 0. 03 06 1v 2 [ m at h. D S] 1 2 O ct 2 02 0 LYAPUNOV EXPONENTS OF PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT ADRIANA SÁNCHEZ AND MARCELO VIANA Abstract. A recent result of Bocker–Viana asserts that the Lyapunov ex- ponents of compactly supported probability distributions in GL(2,R) depend continuously on the distribution. We investigate the general, possibly non- compact case. We prove that the Lyapunov exponents are semi-continuous with respect to the Wasserstein topology, but not with respect to the weak* topology. Moreover, they are not continuous with respect to the Wasserstein topology. 1. Introduction Let M = SL(2,R)Z and f :M →M be the shift map over M defined by (αn)n 7→ (αn+1)n. Consider the function A : M → SL(2,R), (αn)n 7→ α0, and define its n-th iterate by An((αk)k) = αn−1 · · ·α0. Given any probability measure p in SL(2,R), the associated Bernoulli measure µ = pZ on M is invariant under f . Let L1(µ) denote the space of µ-integrable functions on M . Assuming that log+ ‖A±‖ ∈ L1(µ), it follows from Furstenberg- Kesten [5, Theorem 2] that λ+(x) = lim n 1 n log ‖An(x)‖ and λ−(x) = lim n 1 n log ‖A−n(x)‖−1, exist for µ-almost every x ∈M . Furthermore, the functions x 7→ λ±(x) are constant on the orbits of f . Thus, by the ergodicity of the Bernoulli measure, they are constant on a full µ-measure set. We call these constants the Lyapunov exponents of µ, and we represent them as λ±(p). The way Lyapunov exponents depend on the corresponding probability distri- bution p has been studied by several authors. Furstenberg, Kifer [6] proved that this dependence is continuous at every quasi-irreducible point, that is, such that there is at most one subspace invariant under every matrix in the support of p. Date: October 13, 2020. 2010 Mathematics Subject Classification. Primary: 37H15; Secondary: 37A20, 37D25. Key words and phrases. Lyapunov exponents, linear cocycles, Wasserstein topology. Work was partially supported by Fondation Louis D. – Institut de France (project coordinated by M. Viana), CNPq and FAPERJ. A.S. was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), grant 2018/18990-0 and Universidad de Costa Rica. 1 http://arxiv.org/abs/1810.03061v2 2 ADRIANA SÁNCHEZ AND MARCELO VIANA Also under an irreducibility condition, LePage [7] proved Hölder continuity of the Lyapunov exponents. More recently, Bocker and Viana [2] proved that continuity actually holds at every point, in the realm of compactly supported probability distributions. Backes, Butler and Brown [1] have much extended this result, to linear cocycles with invari- ant holonomies, over hyperbolic systems with local product structure on compact spaces. Moreover, still in the compactly supported case, Tall and Viana [10] proved that Hölder continuity holds whenever the exponents λ−(p) and λ+(p) are distinct. In general, when the exponents coincide, one has a weaker but still quantitative, modulus of continuity, called log-Hölder continuity. For much more on this topic, see Viana [9, Chapter 10], Duarte, Klein [4], Viana [8] and the references therein. The aim of this note is to study the (semi)continuity of the Lyapunov exponents in the general case, when the support of the probability measure p is not necessarily compact. Our main result reads as follows (Theorem 3.1 and Theorem 3.2 provide more detailed statements): Theorem A. The function p 7→ λ+(p) is upper semi-continuous for the Wasser- stein topology but not for the weak* topology. The same remains valid for p 7→ λ−(p) with lower semi-continuity. Regarding continuity of Lyapunov exponents we prove (see Theorem 4.1 for a more detailed statement): Theorem B. The functions p 7→ λ±(p) are not continuous for the Wasserstein topology. Acknowledgements. We thank Mauricio Poletti and El Hadji Yaya Tall for many helpful comments and suggestions. 2. Wasserstein Topology Let (M,d) be a Polish space, that is a complete separable metric space. The Wasserstein space is the space of probability measures on M with finite moments of order 1, that is, P1(M) = { µ ∈ P (M) : ∫ M d(x0, x)dµ(x) < +∞ } , where x0 ∈ M is arbitrary and P (M) denotes the space of Borel probability mea- sures on M . This space does not depend on the choice of the point x0. We use µk ∗ −→ µ to mean convergence in the weak* topology. Moreover, µk W −→ µ will mean convergence in the Wasserstein topology, characterized by the following definition (see Definition 6.8 in Villani [11]): Definition 2.1. Let (M,d) be a Polish space and (µk)k∈N be a sequence in P1(M). Given any µ ∈ P1(M), we say that µk W −→ µ if one of the following equivalent conditions is satisfied for some (and then any) x0 ∈M : (1) µk ∗ −→ µ and ∫ d(x0, x)dµk(x) → ∫ d(x0, x)dµ(x); (2) µk ∗ −→ µ and lim sup k→∞ ∫ d(x0, x)dµk(x) ≤ ∫ d(x0, x)dµ(x); PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 3 (3) µk ∗ −→ µ and lim R→∞ lim sup k→∞ ∫ d(x0,x)≥R d(x0, x)dµk(x) = 0; (4) For every continuous function ϕ with |ϕ(x)| ≤ C(1+ d(x0, x)), C ∈ R, one has ∫ ϕ(x)dµk(x) → ∫ ϕ(x)dµ(x). Let (M,µ) and (N, ν) be two probability spaces. A coupling of µ and ν is a measure π on M × N such that π projects to µ and ν on the first and second coordinate, respectively. When µ = ν we call π a self-coupling. If (M,d) is a Polish space, the Wasserstein distance between two probability measures µ and ν on M is defined by W1(µ, ν) = inf π∈Π(µ,ν) ∫ M d(x, y)dπ(x, y), (1) where the infimum is taken over the set Π(µ, ν) of all the couplings of µ and ν. This distance metrizes the Wasserstein topology (see [11, Theorem 6.18]): Theorem 2.1. Let (M,d) be a Polish space. Then the Wasserstein distance W1 metrizes the Wasserstein topology in the space P1(M): µk W −→ µ if and only if W1(µk, µ) → 0. Moreover, with this metric P1(M) is also a complete separable metric space and, every µ ∈ P1(M) can be approximated by probability measures with finite support. This also implies that W1 is continuous on P1(M). 3. Semicontinuity It is a well-known fact that when the measures have compact support, the Lya- punov exponents are semicontinuous with the weak* topology (see for example [9, Chapter 9]). However, in the non compact setting this is no longer true. If they were semicontinuous then every measure with vanishing Lyapunov exponent would be a point of continuity. The next theorem shows that this is not the case. Theorem 3.1. There exist a measure q and a sequence of measures (qn)n on SL(2,R) converging to q in the weak* topology, such that λ+(qn) ≥ 1 for n large enough but λ+(q) = 0. Proof. Define a function α : N → SL(2,R) by α(2k − 1) = ( σk 0 0 σ−1 k ) and α(2k) = ( σ−1 k 0 0 σk ) where (σk)k is an increasing sequence such that σ1 > 1 and σk → +∞. Let µ = qZ be a measure in M where q is the measure on SL(2,R) given by q = ∑ k∈N pkδα(k), with ∑ pk = 1, 0 < pk < 1 for all k ∈ N. The key idea to construct this example is to find pk and σk such that log ‖A‖ ∈ L1(µ) and satisfying the hypothesis above. Hence, consider 0 < r < 1/2 < s < 1, 4 ADRIANA SÁNCHEZ AND MARCELO VIANA and l = s/r > 1. Let us take σk = el k for all k, which is an increasing sequence provided that l > 1. For k ≥ 2 take p2k−1 = p2k = rk. Since 0 < r < 1/2 it is easy to see that ∑ k≥3 pk = 2 ∑ k≥2 p2k = 2 ∑ k≥2 rk = 2 r2 1− r < 1 We have to choose p1 and p2 such that ∑ pk = 1. Then, it is enough to take p1 = p2 = 1 2 ( 1− 2 r2 1− r ) . We continue by showing that log ‖A‖ ∈ L1(µ). This is an easy computation, ∫ M log ‖A‖dµ = 2p2 log σ1 + 2 ∑ k≥2 p2k log σk = 2p2l + 2 ∑ k≥2 sk. Since 0 < s < 1 this geometric series is convergent. More over, since p2k−1 = p2k for all k then λ+(q) = 0. What is left is to construct the sequence qn. Fix n0 > 1 large enough so 1 2 ( 1− 2 r2 1−r ) > l−n for all n ≥ n0, and consider qn = ∑ k q n k δα(k) where for n ≥ n0 qn2n = l−n + rn, qn2 = 1 2 ( 1− r2 1− r ) − l−n qnk = pk other case. Thus, since qnk → pk when n → ∞ for all k, it is easy to see that qn converges in the weak* topology to q. The proof is completed by showing that λ+(qn) ≥ 1 for n large enough. It follows easily since, λ+(qn) = |qn2n−1 − qn2n| log σn + |qn1 − qn2 | log σ1 = l−nln + l−n+1, which is equal to 1 + l−n+1 ≥ 1 for all n ≥ n0. � We now consider the Wasserstein topology in P1(SL(2,R)) which is stronger than the weak* topology, as stated in Definition 2.1. The advantage of using this topology is that all probability measures in P1(SL(2,R)) have finite moment of order 1, and so their Lyapunov exponents always exist. This observation is a direct consequence of the fact that log : [1,∞) → R is a 1-Lipschitz function and ‖α‖ ≥ 1 for every matrix α ∈ SL(2,R), because ∫ log ‖A(x)‖dµ = ∫ log ‖α‖dp ≤ ∫ d(α, id)dp <∞. The convergence of the moments of order 1, allow us to control the weight of integrals outside compact sets, and prove semicontinuity of the Lyapunov exponents in P1(SL(2,R)). We are thus led to the following result: Theorem 3.2. The function defined on P1(SL(2,R)) by p→ λ+(p) is upper semi- continuous. The same remains valid for the function p → λ−(p) with lower semi- continuity. PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 5 Before beginning the proof of Theorem 3.2 we need to recalled some impor- tant results regarding the relationship between Lyapunov exponents and stationary measures. A probability measure η on P 1 is called a p-stationary if η(E) = ∫ η(α−1E)dp(α), for every measurable set E ∈ P 1 and α−1E = {[α−1v] : [v] ∈ E}. Roughly speaking, the following result shows that the set of stationary measures for a measure p is close for the weak* topology. Proposition 3.3. Let (pk)k be probability measures in SL(2,R) converging to p in the weak* topology. For each k, let ηk be pk-stationary measures and ηk converges to η in the weak* topology. Then η is a stationary measure for p. Furthermore, in our context it is well-known that λ+(p) = max { ∫ Φdp× η : η p− stationary } , where Φ : SL(2,R)× P 1 → R is given by Φ(α, [v]) = log ‖αv‖ ‖v‖ . For more details see for example [9, Proposition 6.7]. We now proceed to the proof of Theorem 3.2. Proof of Theorem 3.2. We will prove that λ+(p) is upper semi-continuous. The case of λ−(p) is analogous. Let (pk)k be a sequence in the Wasserstein space P1(M) converging to p, i.e. W (pk, p) → 0. For each k ∈ N let ηk a stationary measure that realizes the maximum in the identity above. That is: λ+(pk) = ∫ Φdpkdηk. Since P 1 is compact, passing to a subsequence if necessary, we can suppose ηk converges in the weak* topology to a measure η which, as established in Proposition 3.3, is a p-stationary measure. Let ǫ > 0, we want to prove that there exist a constant k0 ∈ N such that for each k > k0 ∣ ∣ ∣ ∣ ∫ Φdpkdηk − ∫ Φdpdη ∣ ∣ ∣ ∣ < ǫ. In order to do this we need to consider some properties of the Wasserstein topol- ogy. First of all, since the first moment of p is finite there exist K1 a compact set of SL(2,R) such that ∫ Kc 1 d(α, id)dp < ǫ 36 . (2) Moreover, according to Definition 2.1, since pk W −→ p there exist R′ > 0 satisfying lim sup k ∫ d(α,id)>R′ d(α, id)dpk < ǫ 36 , 6 ADRIANA SÁNCHEZ AND MARCELO VIANA then, there exist k′ > 0 such that for every k > k′ ∫ d(α,id)>R′ d(α, id)dpk < ǫ 36 . (3) Take R > 0 big enough so B(id, R′) ∪ K1 ⊂ B(id, R) and define the compact set K = B̄(id, R). Since the function log : [1,∞) → R is 1-Lipschitz and ‖α‖ ≥ 1 for all α ∈ SL(2,R), then |Φ(α, [v])| = ∣ ∣ ∣ ∣ log ‖αv‖ ‖v‖ ∣ ∣ ∣ ∣ ≤ log ‖α‖ ≤ |‖α‖ − ‖ id ‖| ≤ d(α, id). (4) Our proof starts with the observation that ∣ ∣ ∣ ∣ ∫ Φdpkdηk − ∫ Φdpdη ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∫ K×P1 Φdpkdηk − ∫ K×P1 Φdpdη ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∫ Kc×P1 Φdpkdηk ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∫ Kc×P1 Φdpdη ∣ ∣ ∣ ∣ . From (3) it follows that ∣ ∣ ∣ ∣ ∫ Kc×P1 Φdpkdηk ∣ ∣ ∣ ∣ ≤ ∫ Kc d(α, id)dpk < ǫ 3 . (5) Furthermore, (2) implies that ∣ ∣ ∣ ∣ ∫ Kc×P1 Φdpdη ∣ ∣ ∣ ∣ ≤ ∫ Kc d(α, id)dp < ǫ 3 . (6) We now proceed to analyze the integral: ∣ ∣ ∣ ∣ ∫ K×P1 Φdpkdηk − ∫ K×P1 Φdpdη ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∫ K×P1 Φdpkdηk − ∫ K×P1 Φdpkdη ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∫ K×P1 Φdpkdη − ∫ K×P1 Φdpdη ∣ ∣ ∣ ∣ . Consider ΦK = Φ|K×P1 the restriction of Φ to the compact space K × P 1. Thus, ΦK is uniformly continuous with the product metric. Hence, there exist δ = δ(ǫ) such that for every [v] ∈ P 1 and every α, β ∈ K satisfying d(α, β) < δ we have |ΦK(α, [v])− ΦK(β, [v])| < ǫ 18 . Moreover, by the compactness of the set K we can find α1, ..., αN ∈ K such that K ⊂ ∪N i=1B(αi, δ). Therefore, the convergence of (ηk)k to η in the weak* topology implies that for each i = 1, ..., N there exist ki > 0 such that ∣ ∣ ∣ ∣ ∫ P1 ΦK(αi, [v])dηk − ∫ P1 ΦK(αi, [v])dη ∣ ∣ ∣ ∣ < ǫ 18 . Take k′′ = max{k1, ..., kN}. From the above it follows that given α ∈ K there exist i such that d(α, αi) < δ and for every k > k′′ if Φi = ΦK(αi, [v]) then ∣ ∣ ∣ ∣ ∫ P1 ΦKdηk − ∫ P1 ΦKdη ∣ ∣ ∣ ∣ ≤ ≤ ∫ P1 |ΦK − Φi| dηk + ∣ ∣ ∣ ∣ ∫ P1 Φidηk − ∫ P1 Φidη ∣ ∣ ∣ ∣ + ∫ P1 |Φi − ΦK | dη < ǫ 6 . PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 7 Since this convergence is uniform in α, this implies that ∣ ∣ ∣ ∣ ∫ Kc×P1 ΦK(α, [v])dηkdpk − ∫ Kc×P1 ΦK(α, [v])dηdpk ∣ ∣ ∣ ∣ < ǫ 6 , (7) for all k > k′′. Now define L = SL(2,R)\B(id, R + 1) and, consider the Urysohn function f0 : SL(2,R) → [0, 1] given by f0(α) = d(α,L) d(α,L) + d(α,K) . It is easily seen that f0 is continuous, equal to zero in L and equal to 1 in K. Therefore, the functions ψ(α) = ∫ P1 Φ(α, [v])dη · f0(α) are continuous. Then since |ψ(α)| ≤ d(α, id) and, by the definition of the Wasser- stein topology, there exist k′′′ such that for every k > k′′′ ∣ ∣ ∣ ∣ ∫ ψdpk − ∫ ψdp ∣ ∣ ∣ ∣ < ǫ 18 . Moreover, since ψ − ϕ = 0 in K and |ψ − ϕ| ≤ log ‖α‖|f0(α) − χK(α)| ≤ 2d(α, id). Hence, by (2) and (3) ∫ |ψ − ϕ|dp ≤ 2 ∫ Kc d(α, id)dp < ǫ 18 , ∫ |ψ − ϕ|dpk ≤ 2 ∫ Kc d(α, id)dpk < ǫ 18 for each k > k′. Thus, if k > max{k′, k′′′} we get ∣ ∣ ∣ ∣ ∫ K×P1 Φdpkdη − ∫ K×P1 Φdpdη ∣ ∣ ∣ ∣ < ǫ 6 . (8) Finally, taking k0 = max{k′, k′′, k′′′}, we conclude that for every k > k0 ∣ ∣ ∣ ∣ ∫ Φdpkdηk − ∫ Φdpdη ∣ ∣ ∣ ∣ < ǫ. We just proved that λ+(pk) = ∫ Φdpkdηk → ∫ Φdpdη ≤ λ+(p), which concludes our proof. � Remark 3.4. Theorem 3.1 does not contradicts Theorem 3.2, since p is not in P1(SL(2,R)). To see this, take x0 = id and note that ∫ d(x, x0)dp = ∞ ∑ k=0 pk‖αk − id ‖ = 2 ∞ ∑ k=0 rk(el k − 1), which diverges. 8 ADRIANA SÁNCHEZ AND MARCELO VIANA 4. Proof of Theorem B At this section we are going to describe a construction of points of discontinuity of the Lyapunov exponents as functions of the probability measure, relative to the Wasserstein topology. Theorem 4.1. There exist a measure q and a sequence of measures (qn)n on SL(2,R) converging to q in the Wasserstein topology, such that λ+(qn) = 0 for all n ∈ N but λ+(q) > 0. Proof. Consider the function α : N → SL(2,R) defined by the hyperbolic matrices α(k) = ( k 0 0 k−1 ) Take m ∈ N the smallest natural number bigger than 1 such that ∑ n≥m e− √ n < 1, which exist since ∑ k e − √ k is convergent, and define pk = e− √ k, if k ≥ m, p1 = 1− ∑ n≥m e− √ n, pk = 0, otherwise. It is obvious from the definition that ∑ k pk = 1. Hence, we define the probability measure q = ∑ pkδα(k). We need to see that q ∈ P1(SL(2,R)), in order to do so notice that if x0 = id ∫ d(x, x0)dq = ∑ k pk‖α(k)− id ‖ = ∑ k e− √ k(k − 1) which is convergent by the Cauchy condensation test. Moreover, since ∑ k e− √ k log k < ∑ k ke− √ k then if µ = qZ we have log ‖A‖ ∈ L1(µ) and, λ+(q) = ∑ k e− √ k log k > 0. Now, consider B = ( 0 −1 1 0 ) and for each n consider βn(k) = { α(k) if k 6= n, B if k = n. With this we define the probability measures qn = ∑ k pkδβn(k). In a similar way as above, we can see that for all n these measures belong to P1(SL(2,R)) and, log ‖A‖ ∈ L1(qn). We proceed to show that qn converges to q in the Wasserstein topology. This follows since W (qn, q) ≤ pnd(α(n), B) ∼ ne− √ n which goes to 0 if n goes to ∞. PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 9 It remains to prove that λ+(qn) = 0 for every n. In order to do this we proceed by contradiction. Suppose there exist N such that λ+(qN ) > 0 > λ−(qN ). We will consider the distance in the projective space P 1 given by δ([v], [w]) := ‖v ∧ w‖ ‖v‖‖w‖ = sin(∠(v, w)). Consider the family F = {V,H}, where V = [e2] and H = [e1] are the vertical and horizontal axis respectively. By the definition of the measure qN , it is clear that this family is invariant by every matrix in the support of this measure. Moreover, if x = (xk)k ∈M then we can see that for every m δ(Am(x)H,Am(x)V ) ≥ δ(H,V ) = 1. (9) On the other hand, we have for every unit vectors v and w ‖Am(x)v ∧ Am(x)w‖ ≤ ‖ ∧2 Am(x)‖. It is widely known that, for qN -almost every x ∈M λ+(qN ) + λ−(qN ) = lim m 1 m log ‖ ∧2 Am(x)‖. Note that qN is irreducible, this means that there is no proper subspace of R2 invariant under all the matrices in the support of qN . Therefore, we have λ+(qN ) = lim m 1 m log ‖Am(x)w‖, for every unit vector w. For a more detailed discussion of the two results mention above we refer the reader to [3, Chapter III]. Thus, we have for every unit vectors v and w lim m 1 m log ‖ ∧2 Am(x)‖ ‖Am(x)v‖‖Am(x)w‖ = λ−(qN )− λ+(qN ) < 0, and hence lim m δ(Am(x)H,Am(x)V ) ≤ lim m ‖ ∧2 Am(x)‖ ‖Am(x)e1‖‖Am(x)e2‖ = lim m exp ( m · 1 m log ‖ ∧2 Am(x)‖ ‖Am(x)e1‖‖Am(x)e2‖ ) = 0, which is a contradiction with (9) and, we finish our proof. � Notice that this example shows that the Wasserstein topology is not enough to guarantee continuity of the Lyapunov exponents. The main problem is that the support of the measures qn move further apart from the support of q. Thus, this suggest that we need to add some hypothesis guaranteeing the “convergence” of the supports. An assumption of this type was made by Bocker, Viana in [2] in order to prove the continuity for measures with compact support. However in the non compact setting we will also need to control the iteratives of the cocycles. Those hypotheses are very restrictive so we are left with the following conjecture: Conjecture 4.1. The Lyapunov exponent functions λ± : P1(SL(2,R)) → R are not continuous with respect to the Wasserstein–Hausdorff topology. 10 ADRIANA SÁNCHEZ AND MARCELO VIANA In the next two sections we are going to describe a construction of points of discontinuity of the Lyapunov exponents as functions of the measure, relative to the Wasserstein topology. However, in each of them the support of the measures are arbitrarily close. These constructions were inspired by the discontinuity example presented by Bocker Viana in [2, Section 7.1], and do not prove our conjecture since they are not examples in SL(2,R). 4.1. Discontinuity example in SL(2,R)5. Let us recall that M = (SL(2,R))Z, f :M →M is the shift map over M defined by (αn)n 7→ (αn+1)n. And the linear cocycle A is the product of random matrices which is defined by A : M → SL(2,R), (αn)n 7→ α0. Given an invariant measure p in SL(2,R) we can define µ = pZ which is an invariant measure in M . Now consider X = SL(2,R)5 with the product metric d∞((α1, ..., α5), (β1, ..., β5)) = max{d(α1, β1), ..., d(α5, β5)}. Let N = XZ be the space of sequences over X and g : N → N the shift map over N . We can identify N with M using the function ι : M → N by ι((αn)n) = (βn)n where βn = (α5n, α5n+1, α5n+2, α5n+3, α5n+4)n. It is easy to see that ι defines a bijection between N and M . Moreover, we have the following identity g(ι((αn)n)) = f5((αn)n). Also we can consider the linear cocycle induced by A in N , that is the function B : N → SL(2,R) given by B((ι((αn)n)) = A5((αn)n). So in this context we have the following result. Theorem 4.2. There exist a measure q and a sequence of measures (qn)n on X converging to q in the Wasserstein topology, such that λ+(B, qn) 9 λ+(B, q). The main idea of the proof is to construct a measure on N whose Lyapunov exponents are positive and approximate it, in the Wasserstein topology, by measures with zero Lyapunov exponents. In order to do that, define the function α : N → SL(2,R) as α(2k − 1) = ( k 0 0 k−1 ) and α(2k) = ( k−1 0 0 k ) As in the example before take m ∈ N the smallest natural (odd) number bigger than 3 such that ∑ k≥m e− √ k < 1, which exist since ∑ k e − √ k is convergent, and PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 11 define p2k = p2k−1 = 1 2 e− √ k, if 2k − 1 ≥ m, p3 = 1− ∑ n≥m e− √ n, pk = 0, otherwise. Let µ = q̃Z be a measure in M where q̃ is the measure on SL(2,R) given by q̃ = ∑ k∈N pkδα(k). Let us consider the space Ω = N 5 and define the measure on X by q = ∑ w∈Ω pwδα(w), where α(w) = (α(w1), · · · , α(w5)) and, pw = pw1 · · · pw5 if w = (w1, ..., w5). Now consider the measure ν = qZ on N . First, we need to ensure that the mea- sure q belong to P1(X). This is a direct consequence of the fact that ∑ e− √ n(n−1) is convergent equal to some positive constant c. Indeed, if α0 = (id, ..., id) and the notation p1 · · · p̂i · · · p5 denotes the product of p1 through p5 except pi then ∫ d∞(α, α0)dq = ∑ w pwd∞(α(w), α0) < 5 ∑ i=1 ∑ wj ,j 6=i pw1 · · · p̂wi · · · pw5 ( ∑ wi pwi d(α(wi), id) ) < c 5 ∑ i=1 ∑ wj ,j 6=i pw1 · · · p̂wi · · · pw5 = 5c which proves our claim. Remember that this also guarantees the existence of λ±(B, q) as mention in Section 3. It is easy to see that ν = ι∗µ. Using this we have λ+(B, q) = lim n 1 n ∫ N log ‖Bn(x)‖dν = lim n 1 n ∫ M log ‖A5n(x)‖dµ = 5λ+(A, q̃) = 5p3 log 2 > 0. The task is now to construct the sequence (qn)n. In order to do this, for each n ∈ N consider wn = (2n, 2n+ 2, 2n+ 1, 2n− 1, 2n− 1) and define β(wn) = (α(2n)Rǫ, α(2n+ 2), α(2n+ 1)Rδ, α(2n− 1), α(2n− 1)Rǫ), where ǫ = n−1(n+ 1)−1, δ = arctan(ǫ) and, Rǫ = ( 1 0 ǫ 1 ) , Rδ = ( cos(δ) − sin(δ) sin(δ) cos(δ) ) . We proceed to define the sequence by qn = ∑ w 6=wn pwδα(w) + pwn δβ(wn). 12 ADRIANA SÁNCHEZ AND MARCELO VIANA We claim that W (qn, q) → 0 if n goes to infinite. Our proof starts with the obser- vation that πn = ∑ w 6=wn pwδ(α(w),α(w)) + pwn δ(α(wn),β(wn)) is a coupling of q and qn. Then, W (qn, q) ≤ ∫ d∞(u, v)πk(u, v) = pwn d(α(wn), β(wn)) and the latter is bounded from above by max{‖α(2n)(1−Rǫ)‖, ‖α(2n− 1)(1 −Rδ)‖, ‖α(2n+ 1)(1−Rǫ)‖} ≤ ǫ(n+ 1) = 1 n which proves our claim. What is left is to show that λ+(B, qn) = 0 for all n. For this we need the following lema. Lemma 4.3. Let Hx = R(1, 0) and Vx = R(0, 1). If Zn = [0 : β(wn)] then, for all x ∈ Zn we have B(x)Hx = Vg(x) and B(x)Vx = Hg(x) Proof. Notice that for any x ∈ Zn B(x) = ( 0 −ǫ−2 sin(δ) ǫ2 sin(δ) + ǫ cos(δ) 0 ) . Which completes the proof. � The rest of the proof follows the same arguments as the Bocker-Viana example. For more details we refer the reader to ([2, Section 7.1]). 4.2. Discontinuity example in GL(2,R)2. LetM = (GL(2,R))Z let f :M →M be the shift map over M and A : M → GL(2,R) the product of random matrices. Now consider X = GL(2,R)2 with the maximum norm, and let N = XZ be the space of sequences over X and g : N → N the shift map over N . As before, we can identify N with M using the function ι :M → N defined by ι((αn)n) = (βn)n where βn = (α2n, α2n+1) which is a bijection between N and M . With the above definition we can see that g(ι((αn)n)) = f2((αn)n) and defined B : N → GL(2,R) the linear cocycle induced by A in N through B(ι((αn)n)) = A2((αn)n). In a similar way as in the previous example, there exist a measure p and a sequence of measures (pk)k on X converging to p in the Wasserstein topology, such that λ+(A, pk) 9 λ+(A, p). Indeed, let α : N → GL(2,R) be defined by α(2k − 1) = ( k 0 0 k−2 ) and α(2k) = ( k−2 0 0 k ) . Take the weights pk as in the previous section and let q̃ = ∑ k∈N pkδα(k), . Consider the space Ω = N 2 and define the measure on X by q = ∑ w∈Ω pwδα(w), where α(w) = (α(w1), α(w2)) and, pw = pw1 pw2 if w = (w1, w2). Let ν = qZ a measure on N . PROBABILITY DISTRIBUTIONS WITH NON-COMPACT SUPPORT 13 Analysis similar to that in Section 4.1 shows that q ∈ P1(X), and using that ν = i∗µ we have λ+(B, q) = lim n 1 n ∫ N log ‖Bn(x)‖dν = lim n 1 n ∫ M log ‖A2n(x)‖dµ = 2λ+(A, q̃) = 2p3 log 2 > 0. For each n ∈ N consider wn = (2n, 2n− 1) and define β(wn) = (β(2n), β(2n− 1)), by β(2n) = ( 1 −δ 0 1 ) α(2n) ( 1 0 ǫ 1 ) = ( 0 −nδ ǫn n ) , β(2n− 1) = ( 1 0 ǫ 1 ) α(2n− 1) = ( n 0 ǫn n−1 ) , where δ = n−(1+γ) with 0 < γ < 1, ǫ = n−3δ−1 = nγ−2. We proceed to define the sequence by qn = ∑ w 6=wn pwδα(w) + pwn δβ(wn). To prove that W (qn, q) → 0 if n goes to infinite we consider the diagonal coupling of qn and q πn = ∑ w 6=wn pwδ(α(w),α(w)) + pwn δ(α(wn),β(wn)) Hence, we have W (qn, q) ≤ ∫ d∞(u, v)πn(u, v) = pwn d∞(α(wn), β(wn)) < max{‖β(2n)− α(2n)‖, ‖β(2n− 1)− α(2n− 1)‖} ≤ max{ǫσn, n −2 + nδ} = max{nγ−1, n−2 + n−γ} ≤ 2n−l where l = min{γ, 1− γ} > 0, which proves our claim. The rest of the proof, that is proving that λ+(B, qn) = 0 for all n, runs as before by noticing that for any x ∈ Zn = [0 : β(wn)] B(x) = ( 0 −n2δ ǫn−1 0 ) . Indeed, this guarantees that B(x)Hx = Vg(x) and B(x)Vx = Hg(x) where Hx = R(1, 0) and Vx = R(0, 1). Finally, applying the first return map argument in [2, Section 7.1], we conclude our proof. Acknowledgements The first author thanks the Math Department of ICMC where the major part of the work was developed. This work is supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), grant 2018/18990-0. 14 ADRIANA SÁNCHEZ AND MARCELO VIANA References [1] Lucas Backes, Aaron W Brown, and Clark Butler. Continuity of lyapunov exponents for cocycles with invariant holonomies. J. Mod. Dyn., 12:223–260, 2018. [2] Carlos Bocker-Neto and Marcelo Viana. Continuity of lyapunov exponents for random two- dimensional matrices. Ergodic Theory and Dynamical Systems, 37(5):1413–1442, 2017. [3] Philippe Bougerol et al. Products of random matrices with applications to Schrödinger oper- ators, volume 8. Springer Science & Business Media, 2012. [4] P. Duarte and S. Klein. Lyapunov exponents of linear cocycles, volume 3 of Atlantis Studies in Dynamical Systems. Atlantis Press, 2016. Continuity via large deviations. [5] H. Furstenberg and H. Kesten. Products of random matrices. Ann. Math. Statist., 31:457–469, 1960. [6] H. Furstenberg and Yu. Kifer. Random matrix products and measures in projective spaces. Israel J. Math, 10:12–32, 1983. [7] É. Le Page. Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Statist., 25:109– 142, 1989. [8] M. Viana. (Dis)continuity of Lyapunov exponents. Ergod. Theory Dynam. Systems. [9] Marcelo Viana. Lectures on Lyapunov Exponents, volume 145. Cambridge University Press, 2014. [10] Marcelo Viana and El Hadji Y. Tall. Moduli of continuity of lyapunov exponents for random gl(2) cocycles. final preparation, 2018. [11] Cédric Villani. Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008. ICMC-USP – Av. Trab. São-carlense 400, São Carlos, 13566-590 São Paulo, Brazil. Email address: asanchez@icmc.usp.br IMPA – Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil. Email address: viana@impa.br 1. Introduction 2. Wasserstein Topology 3. Semicontinuity 4. Proof of Theorem B 4.1. Discontinuity example in SL(2,`39`42`"613A``45`47`"603AR)5 4.2. Discontinuity example in GL(2,`39`42`"613A``45`47`"603AR)2 Acknowledgements References