M¨ obius transformations and Furstenberg’s theorem Piotr Rutkowski BSc Wednesday 8 th April, 2020 M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 1/29
Theorem 1 - discrete version Theorem Let F = { f 1 , . . . , f N ; p 1 , . . . , p N } , p j > 0, ∑ N 1 p j = 1 be an IFS (iterated function system) with probabilities of M¨ obius transformations f i mapping the upper half-plane H onto itself, and having no common invariant (hyperbolic) line in H . Then, for any initial point Z 0 ∈ H , the orbit Z n = Z n ( Z 0 ) = f n ◦ f n − 1 ◦ · · · ◦ f 1 ( Z 0 ) , n ≥ 1 , tends to R almost surely, as n → ∞ . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 2/29
Theorem 1 Theorem Let µ be a probability measure on the set of M¨ obius transformations mapping the upper half-plane H = { Im z > 0 } onto itself. Assume that the transformations in the support of µ have no common fixed point in H and no common invariant (hyperbolic) line in H . Let { F n } be iid random variables with distribution µ . Then, for any initial point Z 0 ∈ H , the orbit { Z n } ∞ 0 tends to R almost surely. That is, for arbitrarily fixed compact subset K ⊂ H and for almost all orbits { Z n } n ≥ 1 , only a finite number of points of the orbit belong to K. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 3/29
Furstenberg’s theorem Theorem Let µ be a probability measure on SL ( 2, R ) , such that the following holds (if G µ is the smallest closed subgroup of SL ( 2, R ) which contains the support of µ ) 1 G µ is not compact; 2 there does not exist a subset L of R 2 which is a finite union of one-dimensional subspaces, such that M ( L ) = L for any M in G µ . Then, with probability 1, the norms � Y n . . . Y 1 x � grow exponentially as n → ∞ , for all x ∈ R 2 \{ 0 } , where { Y n } are iid random variables with values in SL ( 2, R ) and distribution µ . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 4/29
Theorem 2 Theorem Let µ be a probability measure on SL ( 2, R ) . Assume that the matrices in supp µ have no common invariant elipse, and no common invariant set of type l 1 ∪ l 2 , with lines l 1 , l 2 (not necessarily different) passing through the origin. Then, with probability 1 , the norms � Y n . . . Y 1 x � grow exponentially, as n → ∞ , for all x ∈ R 2 \{ 0 } , where { Y n } are iid random variables with values in SL ( 2, R ) and distribution µ . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 5/29
Lemma 1 Lemma � a � b Let { A = } A be a subset of SL ( 2, R ) and let c d { f A ( z ) = az + b cz + d } A be the associated M¨ obius transformations. Then, for any fixed z ∈ H , { f A ( z ) } A is an unbounded set in H (in the hyperbolic sense) if and only if the norms {� A �} A are unbounded (in R with the ordinary Euclidean metric). M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 6/29
Proof of Lemma 1 - part 1/3 M¨ obius transformations preserve hyperbolic distances. Hence, instead of considering general point z ∈ H , we can consider the point z = i . Then f A ( i ) = ai + b ci + d = ( bd + ac ) + i ( ad − bc ) c 2 + d 2 . Since 1 = detA = ad − bc , we get f A ( i ) = bd + ac 1 c 2 + d 2 + i c 2 + d 2 . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 7/29
Proof of Lemma 1 - part 2/3 Now let’s assume that { f A ( i ) } A is an unbounded set in H in hyperbolic sense. This gives us three following alternatives: 1 { c 2 + d 2 } A is unbounded (in R ), 2 { bd + ac c 2 + d 2 } A is unbounded, 3 { c 2 + d 2 } A contains element arbitrarily close to 0. All these alternatives together with detA = ad − bc = 1 assure us that {| a | + | b | + | c | + | d |} A = {� A �} A is unbounded in R . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 8/29
Proof of Lemma 1 - part 3/3 Now, conversely, {� A �} A is unbounded. If, in addition, {| c | + | d |} A is unbounded, { f A ( i ) } A has element arbitrarily close to real line. If {| c | + | d |} A is bounded, {| a | + | b |} A is unbounded. In this case √ a 2 + b 2 | f A ( i ) | = | ai + b √ ci + d | = c 2 + d 2 and | f A ( i ) | A is unbounded in R . In either case the set { f A ( i ) } A is an unbounded subset of H . Lemma 1 is proved. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 9/29
Lemma 2 Lemma In the notation of Lemma 1 , the matrices { A } A have a common invariant ellipse if and only if the associated M¨ obius transformations { f A } A have a common fixed point in H . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 10/29
Proof of Lemma 2 - part 1/2 obius transformations { f A } A have a common Let the set of M¨ fixed point w ∈ H . Choose B ∈ SL ( 2, R ) , such that for the obius transformation f B we have f B ( i ) = w . associated M¨ Then i is a common fixed point of the conjugate system { f − 1 ◦ f A ◦ f B } A . B M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 11/29
Proof of Lemma 2 - part 2/2 We should also note that f − 1 ◦ f A ◦ f B = f B − 1 AB and i is a fixed B point of f ( z ) = az + b cz + d iff a = d and b = − c . � a � � a � b b If, in addition, det ( ) = 1, is a rotation around c d c d the origin. This shows that { B − 1 AB } A have a common invariant circle centered at the origin in R 2 . Then, { A } A has a common invariant ellipse which is the image of the unit circle under B . The converse is proved in the same manner. Hence, Lemma 2 is proved. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 12/29
Lemma 3 Lemma Let µ be a probability measure on the set of M¨ obius transformations mapping the upper half-plane H onto itself. Assume that the transformations in the support of µ have no common fixed point in H . Then for any initial fixed point z ∈ H the set { f ( z ) } , f ∈ G µ , is unbounded (in the hyperbolic sense), where G µ denotes the smallest subgroup of M¨ obius transformations containing the support of µ . Proof: too long. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 13/29
Proposition 1 Proposition Let µ be a probability measure on SL ( 2, R ) and let G µ be the smallest closed subgroup of SL ( 2, R ) which contains the support of µ . Then G µ is compact iff all the matrices in G µ (or equivalently in supp µ ) have a common invariant ellipse. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 14/29
Proof of Proposition 1 - part 1/2 In one direction the assertion is evident: if all matrices in G µ have a common invariant ellipse, then the group G µ is compact. Now assume that the matrices in G µ have no common invariant ellipse. Due to Lemma 2, this is equivalent to the fact that the obius transformations f A , where A ∈ G µ , have no associated M¨ common fixed point in H . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 15/29
Proof of Proposition 1 - part 2/2 Next we use Lemma 3, which says that in this case the set { f A ( z ) } A is unbounded (in hyperbolic sense) in H , for any fixed z ∈ H . Finally, Lemma 1 shows that {� A �} A is unbounded and, hence, that G µ is non-compact, which finishes the proof. Proposition 1 is proved. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 16/29
Proposition 2 Proposition Assume that the group G µ in Proposition 1 is non-compact and that there exists a subset L = ∪ l i of R 2 of a finite union of n different one-dimensional subspaces l i , i = 1, . . . , n, such that A ( L ) = L for all A ∈ G µ . Then n ≤ 2 . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 17/29
Proof of Proposition 2 - part 1/2 Let ∆ be a unit disc in R 2 . The lines l i , where i = 1, . . . , n , intersect at the origin and divide ∆ into 2 n parts. Let’s denote them by D i , where i = 1, . . . , i = 2 n . Let A ∈ G µ . A ( ∆ ) is an ellipse centered at the origin. We also know that A ( L ) = L . The set L divides A ( ∆ ) into 2 n parts, which we’ll denote as S i , for i = 1, . . . , 2 n . Now, Proposition 2 will be proved by showing that if n > 2, at least one S i will have an arbitrarily small area, provided that the norm of A is sufficiently large. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 18/29
Proof of Proposition 2 - part 2/2 First we assume that n = 3 ( L = l 1 ∪ l 2 ∪ l 3 ). If � A � is large, A ( ∆ ) is a long and thin origin-centered ellipse with unit area. Let’s denote by P 1 , P 2 points of the ellipse having maximal distance from each other, and denote by l a straight line passing through these points. Then l passes through the origin as well. Let l 1 form the minimal angle with l among all l i . Then the parts S i of A ( ∆ ) lying between the lines l 2 and l 3 and not containing points P i will have arbitrarily small area if A ( ∆ ) is sufficiently long and thin. The proof proceeds in similar fashion for n > 3. Proposition 2 is proved. M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 19/29
Theorem 2 Theorem Let µ be a probability measure on SL ( 2, R ) . Assume that the matrices in supp µ have no common invariant elipse, and no common invariant set of type l 1 ∪ l 2 , with lines l 1 , l 2 (not necessarily different) passing through the origin. Then, with probability 1 , the norms � Y n . . . Y 1 x � grow exponentially, as n → ∞ , for all x ∈ R 2 \{ 0 } , where { Y n } are iid random variables with values in SL ( 2, R ) and distribution µ . M ¨ OBIUS TRANSFORMATIONS AND F URSTENBERG ’ S THEOREM 20/29
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