A Central Limit Theorem for Rosen Continued Fractions
aa r X i v : . [ m a t h . D S ] S e p A CENTRAL LIMIT THEOREM FORROSEN CONTINUED FRACTIONS
Juno Kim, Kyuhyeon ChoiSeptember 8, 2020
Abstract
We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. ALasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded vari-ation space, which implies quasi-compactness of the transfer operator. The main result is a direct proofof the existence of a spectral gap, assuming a certain behavior of the transformation when iterated. Thiscondition is explicitly proved for the Rosen system. We conclude via well-known results of A. Broise thatthe central limit theorem holds.
Our main systems of interest are the
Rosen continued fractions , a generalization of the Euclidean nearest-integer algorithm. For a fixed integer q ≥
3, let λ q = 2 cos( πq ) and I q = [ − λ q / , λ q / T q : I q → I q is given as: T q ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) x (cid:12)(cid:12)(cid:12)(cid:12) − λ q (cid:22)(cid:12)(cid:12)(cid:12)(cid:12) λ q x (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:23) , T (0) = 0 . This corresponds to a continued fraction expansion of the form x = ǫ a λ q + ǫ a λ q + ··· , (1)where ǫ n = sgn( T n − ( x )) and a n = (cid:4)(cid:12)(cid:12) /λ q T n − x (cid:12)(cid:12) + 1 / (cid:5) . When q = 3, we retrieve the nearest-integeralgorithm.Our goal is to show that Birkhoff sums of the Rosen continued fractions are asymptotically Gaussian. In[2], Broise showed such a central limit system for general piecewise expanding maps of the interval under someconditions, including the assumption that 1 is a simple eigenvalue of the corresponding Perron-Frobeniusoperator. Our paper gives a proof of the existence of a spectral gap for Rosen continued fractions.We develop our theory for a more general class of 1-dimensional continued fractions, detailed in Section2. In Section 3, we show a Lasota-Yorke type bound, which is used in Section 4 to show quasi-compactnessof the transfer operator. Next, in Section 5, this result is strengthened to prove the existence of a spectralgap under a certain technical hypothesis concerning the transformation. This hypothesis will be shown tohold for the Rosen continued fractions in Section 6. Finally, we conclude with the statement of the CentralLimit Theorem in Section 7. The results of Sections 3 and 4 hold in general for a wide class of expanding maps of intervals. One suitablybroad framework in which we may work in is the theory of
Iwasawa continued fractions , developed in detailin [3]. This includes the Euclidean algorithm, the even, odd and centered algorithms, continued fractions,alpha continued fractions, Rosen continued fractions, the alpha-Rosen extensions, and their folded variants.In this framework, T is a map from the interval I = [ a, b ] to itself, which admits a countable partitioninto subintervals such that T is of the form ι ( x )+(constant) on each subinterval. Here, ι denotes one of the1nversions ± /x , ± | /x | , so that | T ′ ( x ) | = 1 /x for all x ∈ I . We denote by H the set of all inverse branchesof T , and require that every branch has domain the full interval I , except possibly the leftmost and rightmostbranches, whose domain should still share an endpoint with I .The continued fraction is called proper if γ := max {| a | , | b |} <
1; we will require this assumption through-out this paper. The only significant algorithm which is not proper is the standard Euclidean algorithm,whose normality have already been studied (see [8]).
Lemma 1.
For a proper one-dimensional Iwasawa continued fraction, the following bounds hold for all n : sup I | ( T n ) ′ | ≤ γ n and sup I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) | ( T n ) ′ | (cid:19) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ γ − γ n − γ Proof.
For the first inequality, 1 | ( T n ) ′ ( x ) | = 1 Q n − i =0 | T ′ ( T i ( x )) | = n − Y i =0 T i ( x ) ≤ γ n . For the second inequality, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) | ( T n ) ′ | (cid:19) ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − Y i =0 T i ( x ) ! ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n − X i =0 (cid:12)(cid:12) T i ( x ) (cid:12)(cid:12) i − Y j =0 T ′ ( T j ( x )) n − Y j =0 , j = i T j ( x ) = 2 n − X i =0 (cid:12)(cid:12) T i ( x ) (cid:12)(cid:12) n − Y j = i +1 T j ( x ) ≤ n − X i =0 γ n − i − = 2 γ − γ n − γ , where an empty product is defined to be 1.The transfer operator H on L ( I ) is given, by an application of the change-of-variables formula, as follows: Hf ( x ) = X h ∈H | h ′ ( x ) | f ◦ h ( x ) · χ T I h ( x )Note that k Hf k ≤ R I P h ∈H | h ′ ( x ) | | f ◦ h ( x ) | dx = R I H | f | ( x ) dx = R I | f ( x ) | dx , hence k H k ≤
1. Infact, there exists an invariant probability measure f m for T by [1], where m denotes Lebesgue measure on I . Since f must satisfy Hf = f , we have k H k = 1.Letting H n denote the set of inverse branches of T n , we more generally have: H n f ( x ) = X h ∈H n | h ′ ( x ) | f ◦ h ( x ) · χ T n I h ( x ) . For each n ∈ N , { I h : h ∈ H n } is a countable partition of I (up to overlapping endpoints). Some simplebookkeeping of subintervals will yield the following. Proposition 1.
The set { T n I h : h ∈ H n } is finite, and consists only of the full interval I or intervals sharingan endpoint with I .In particular, the other endpoint must be one of T i ( a ) or T i ( b ) for 0 ≤ i ≤ n , hence the finiteness. Corollary 1.
For each n , there exists an ǫ n > such that m ( T n I h ) ≥ ǫ n for all h ∈ H n . In this section, we derive a Lasota-Yorke type inequality using the space of bounded variation. We first statesome well-known facts about bounded variation. In order to apply Theorem 1, we work with complex-valuedfunctions. 2ecall that a function f on a closed interval J is of bounded variation on J if:var J f := sup P | P | X i =1 | f ( x i ) − f ( x i − ) | < ∞ , where the supremum runs over all partitions P = { min J = x < x < · · · < x n = max J } .The bounded variation space BV ℓ ( J ) is the Banach space consisting of all left-continuous, integrablefunctions of bounded variation on J with the norm, k f k BV := Z J | f ( x ) | dx + var J f Taking each element of BV ℓ ( J ) as its equivalence class modulo null sets, we may view the boundedvariation space as a subspace BV ( J ) of L ( J ), with the variation now being taken as the infimum over allequivalent functions (see for example [6]): k f k BV = k f k L ( J ) + inf g = f a . e . var J g .For f, g ∈ BV ( J ), we have the following basic inequalities:var J f g ≤ sup J | f | var J g + sup J | g | var J f , var J f g ≤ sup J | g | var J f + sup J | g ′ | Z J | f ( x ) | dx if g ∈ C ( J ) , sup J | f | ≤ var J f + 1 m ( J ) Z J | f ( x ) | dx, . In addition, if J ⊆ I and J shares an endpoint with I , var I f χ J ≤ var J f + sup J | f | .We now evaluate var I H n f :var I H n f = var I X h ∈H n | h ′ | f ◦ h · χ T n I h ! ≤ X h ∈H n var I ( | h ′ | f ◦ h · χ T n I h ) ≤ X h ∈H n var T n I h ( | h ′ | f ◦ h ) + X h ∈H n sup T n I h || h ′ | f ◦ h | = X h ∈H n var T n I h (cid:18) f ◦ h | ( T n ) ′ ◦ h | (cid:19) + X h ∈H n sup T n I h (cid:12)(cid:12)(cid:12)(cid:12) f ◦ h ( T n ) ′ ◦ h (cid:12)(cid:12)(cid:12)(cid:12) ≤ X h ∈H n var T n I h (cid:18) f ◦ h | ( T n ) ′ ◦ h | (cid:19) + X h ∈H n m ( T n I h ) Z T n I h (cid:12)(cid:12)(cid:12)(cid:12) f ◦ h ( T n ) ′ ◦ h (cid:12)(cid:12)(cid:12)(cid:12) dx = 2 X h ∈H n var I h (cid:18) f | ( T n ) ′ | (cid:19) + X h ∈H n m ( T n I h ) Z I h | f ( x ) | dx ≤ X h ∈H n sup I | ( T n ) ′ | var I h f + X h ∈H n sup I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) | ( T n ) ′ | (cid:19) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 m ( T n I h ) ! Z I h | f ( x ) | dx ≤ γ n var I f + (cid:18) γ − γ n − γ + 1 ǫ n (cid:19) Z I | f ( x ) | dx (2)Thus, fixing k ∈ N large so that 0 < ρ := 2 γ k <
1, we have proven the Lasota-Yorke inequality:var I H k f ≤ ρ var I f + M k f k for some large M . Since k H k = 1, it immediately follows that (cid:13)(cid:13) H k f (cid:13)(cid:13) BV ≤ ρ k f k BV + M k f k (3)where M = M + 1 − ρ . 3 The Ionescu-Tulcea and Marinescu Theorem
To obtain quasi-compactness of the operator H , we refer to the following theorem of Ionescu-Tulcea andMarinescu [4]. Theorem 1.
Let ( B, |·| ) and ( L, k·k ) be two complex Banach spaces with L ⊂ B . Let U be an operator from L to L , bounded with respect to both |·| and k·k , which satisfies the additional conditions:(a) if f n ∈ L , f ∈ B with | f n − f | → and k f n k ≤ M for all n , then f ∈ L and k f k ≤ M ;(b) sup n | U n | L < ∞ ;(c) there exist k ≥ , < ρ < , and M such that (cid:13)(cid:13) U k f (cid:13)(cid:13) ≤ ρ k f k + M k f k ;(d) for any bounded subset L ′ of ( L, k·k ) , U k L ′ has compact closure in ( B, |·| ) .Then the set G of eigenvalues λ of U of modulus 1 is finite, the corresponding eigenspaces E λ are finite-dimensional, and U n can be represented by bounded linear operators U λ and V as: U n = X λ ∈ G λ n U λ + V n (4) such that U λ = U λ , U λ U λ ′ = 0 if λ ′ = λ, U λ V = V U λ = 0 , U λ L = E λ , (5) and V has spectral radius strictly less than 1. We claim that by taking the two spaces to be ( L ( I ) , k·k ) and ( BV ( I ) , k·k BV ) and putting U = H , thefour conditions (a) to (d) are satisfied and H admits a spectral decomposition of the above form. Proof of Claim. (a) follows from the fact that the ball A = { f ∈ BV ( I ) : k f k BV ≤ M } is L -compact;see [7]. Similarly, since the image H k L ′ of a bounded subset L ′ of BV ( I ) is bounded w.r.t. k·k BV , it has L -compact closure, hence (d). For (b), sup n k H n k ≤ sup n k H k n ≤
1. The Lasota-Yorke bound (c) wasshown in the preceding section.
Recall that the operator H preserves the invariant density f of the transformation T , which is of boundedvariation; thus 1 is an eigenvalue of H . In this section, we prove that 1 is a simple eigenvalue, and thereare no other complex eigenvalues of modulus 1. Together with quasi-compactness, this implies that H hasspectral gap. We will show this under the following assumption ( ∗ ) on T , which we verify for the Rosencontinued fractions in Section 6. Condition ( ∗ ) . For any subintervals
J, K of I , there exists N ≥ m ( T N ( I ) ∩ T N ( J )) > Lemma 2.
There exist positive constants C , D such that for all n ≥ and ≤ r < k , (cid:13)(cid:13) H nk f (cid:13)(cid:13) BV ≤ ρ n k f k BV + C k f k , k H r f k BV ≤ D k f k BV . Proof.
By repeated application of the Lasota-Yorke bound, (cid:13)(cid:13) H nk f (cid:13)(cid:13) BV ≤ ρ (cid:13)(cid:13)(cid:13) H ( n − k f (cid:13)(cid:13)(cid:13) BV + M k f k ≤ ρ (cid:16) ρ (cid:13)(cid:13)(cid:13) H ( n − k f (cid:13)(cid:13)(cid:13) BV + M k f k (cid:17) + M k f k ≤ · · · ≤ ρ n k f k BV + M − ρ k f k In addition, by (2), k H r f k BV ≤ γ r var I f + (cid:18) γ − γ r − γ + 1 ǫ r (cid:19) k f k ≤ (cid:18) γ − γ + max ≤ j For f ∈ BV with R I f dm = 0, H n f converges to 0 in L -sense as n → ∞ . Proof. It suffices to prove this statement for real-valued f . Suppose k f k BV ≤ 1. Any positive integer m canbe expressed as m = nk + r with n ≥ 0, 0 ≤ r < k . Then, by the inequalities of Lemma 2, k H m f k BV = (cid:13)(cid:13) H nk + r f (cid:13)(cid:13) BV ≤ ρ n k H r f k BV + C k H r f k ≤ ( C + D ) k f k BV . Since the set { f : k f k BV ≤ C + D } is L -compact, there exists a convergent (in L -sense) subsequence H n s f whose limit is denoted by f ∈ BV ( I ).Now we show that k H n f k = k f k for all positive integers n . Fix ǫ > 0. For all large s , k H n s f − f k < ǫ ,so that k H n f k ≥ k H n + n s f k − ǫ . Fixing t > s such that n + n s < n t , (cid:13)(cid:13) H n + n s f (cid:13)(cid:13) ≥ k H n t f k ≥ k f k − ǫ. Thus k H n f k ≥ k f k − ǫ ; letting ǫ tend to 0, we have k H n f k ≥ k f k for all n . The opposite inequalityis trivial.Since R I H n s f dm = R I f dm = 0, we have R I f dm = 0. Suppose f = 0 a.e. Viewing f as a left-continuousfunction in BV ℓ ( I ), there exists a < α, β < b such that f ( α ) > , f ( β ) < 0, which implies f is strictlypositive (resp. negative) on [ α − ǫ, α ] (resp. [ β − ǫ, β ]), for some ǫ > f is positive on J ⊆ I , then Hf is strictly positive on T ( J ). By Condition ( ∗ ), there exists N such that A = T N ([ α − ǫ, α ]) ∩ T N ([ β − ǫ, β ]) has positive measure.Decompose f into its positive and negative parts, f = f + − f − . Then, (cid:13)(cid:13) H N f + (cid:13)(cid:13) + (cid:13)(cid:13) H N f − (cid:13)(cid:13) = k f k = (cid:13)(cid:13) H N f (cid:13)(cid:13) = (cid:13)(cid:13) H N f + − H N f − (cid:13)(cid:13) . However, both H N f + and H N f − are strictly positive on A , so the triangle inequality must be strict; acontradiction. We conclude that f = 0 a.e.We are now in a position to prove the existence of a spectral gap for H . Theorem 2. The eigenvalue 1 is simple, and there are no other eigenvalues of modulus 1.Proof. Suppose that f is another eigenfunction corresponding to 1. Let g = f R I f dm − f . Then Hg = g and R I gdm = 0. By the preceding proposition, k g k = k H n g k → 0, so g = 0 a.e. Thus f must be a constantmultiple of f .It remains to show that there is no other eigenvalue on the unit circle. Suppose that λ = 1, | λ | = 1 is aneigenvalue of H , with a corresponding eigenfunction g . Since Z I gdm = Z I Hgdm = λ Z I gdm, we have R I gdm = 0. Therefore H n g = λ n g must converge to 0 by the preceding proposition, contradicting | λ | = 1. ( ∗ ) for the Rosen Map In this section, we demonstrate Condition ( ∗ ) for the Rosen continued fractions. In fact, it follows triviallyfrom the theorem below. Theorem 3. For any subinterval [ c, d ] ⊆ [ − λ q / , λ q / , T Nq ([ c, d ]) = [ − λ q / , λ q / a.e. for large enough N .Proof. We prove the theorem when q ≥ 5; the cases q = 3 , σ q = λ q / 2. The ranges I j of the inverse branches h j are, with appropriate indexing: I = [ 13 σ q , σ q ] , I n +1 = [ 1(2 n + 3) σ q , n + 1) σ q ] , I n = − I n − ( n ≥ . Note that √ ∈ I and T q ( √ ) = √ − λ q < 0, so we may choose ǫ > √ − ǫ ∈ I and T q ( √ − ǫ ) < c, d ] exists. Clearly 0 / ∈ [ c, d ], for otherwise we may take N = 1.If 0 < c < d , let c ∈ I n +1 , d ∈ I m +1 ( n ≥ m ). If n > m + 1, T q [ c, d ] must be the full interval, acontradiction. Thus n = m or m + 1. Similarly, if c < d < 0, let c ∈ I n , d ∈ I m , where n = m or m − T q [ c, d ] are given below.type (a) : T q [ c, d ] = [ T q ( d ) , T q ( c )] (0 < c < d, n = m )type (b) : T q [ c, d ] = [ T q ( c ) , T q ( d )] ( c < d < , n = m )type (c) : T q [ c, d ] = [ − σ q , T q ( c )] ∪ [ T q ( d ) , σ q ] (0 < c < d, n = m + 1)type (d) : T q [ c, d ] = [ − σ q , T q ( d )] ∪ [ T q ( c ) , σ q ] ( c < d < , n = m − d ≥ √ − ǫ , T q ( d ) < ∈ T q [ c, d ]. Therefore T q [ c, d ] = [ − λ q / , λ q / < c < d < √ − ǫ . Similarly, − √ + ǫ < c < d < p n , q n ] by [ p , q ] = [ c, d ], and[ p n +1 , q n +1 ] =[ T q ( q n ) , T q ( p n )] if [ p n , q n ] is of type (a)[ T q ( p n ) , T q ( q n )] if [ p n , q n ] is of type (b)[ − T q ( c ) , σ q ] ∪ [ T q ( d ) , σ q ] if [ p n , q n ] is of type (c)[ − T q ( d ) , σ q ] ∪ [ T q ( c ) , σ q ] if [ p n , q n ] is of type (d)This sequence is well defined since T q ( I ) = T q ( − I ) for all I ⊂ [ − λ q / , λ q / η > m ([ p n +1 , q n +1 ]) > η · m ([ p n , q n ]) for all n , which yieldsthe desired contradiction as n → ∞ .Since (cid:12)(cid:12) T ′ q ( x ) (cid:12)(cid:12) = 1 /x , for [ p n , q n ] of type (a) or (b), m ([ p n +1 , q n +1 ]) ≥ σ q m ([ p n , q n ]) . For [ p n , q n ] of type (c) or (d), since max {| p n | , | q n |} < √ − ǫ , | T ′ q ( x ) | ≥ / √ − ǫ ) ∀ x ∈ [ p n , q n ]Therefore, for type (c), m ([ p n +1 , q n +1 ]) ≥ 12 ( m ([ − σ q , T q ( c )]) + m ([ T q ( d ) , σ q ])) ≥ 12 1(1 / √ − ǫ ) m ([ p n , q n ]) . And the same holds for type (d). 6 Central Limit Theorem For the statement of the Central Limit Theorem for operators with a spectral gap, we refer to [2]. For f ∈ BV ( I ), let S N f denote the Birkhoff sum P N − k =0 f ◦ T k , and let µ = f m be the invariant measure of T on I .Following Broise, we define a condition on functions f of bounded variation, which is equivalent to adegeneracy σ = 0 of the variance of the distribution of Birkhoff sums. Condition (H). There exists u ∈ L ( µ ) with f = u − u ◦ T + R I f dµ . Theorem 4. For f ∈ BV ( I ) that does not satisfy Condition (H), there exists σ > so that: lim N →∞ µ (cid:20) S N f − N R I f dµσ √ N ≤ v (cid:21) = 1 √ π Z v −∞ e − t / dt for all v. In particular, by taking f to be constant on each subinterval I h , we may interpret f as a cost functionon the set of inverse branches H . (The orbits which hit the endpoints of I h ’s have measure zero and may beignored.) The condition f ∈ BV ( I ) is satisfied, for example, by monotonic and bounded costs with respectto the integer values a i in the expansion (1). In this case, Condition (H) is satisfied only if f is constant onall branches excluding the leftmost one, as all other branches admit fixed points. Corollary 2. For any nonconstant, monotonic, and bounded cost on the digits | a i | , the total cost of anexecution of the Rosen algorithm, with input randomly chosen from the invariant distribution µ , will beasymptotically Gaussian. Further analyses into Condition (H) for general f are also given in [2]. For example, suppose f is piecewisecontinuous. We may add a constant to f so that R I f dµ = 0. Now, if a point x ∈ I with period k can be foundwhose orbit does not meet the countable set consisting of the orbits of discontinuities of f or the endpointsof the subintervals I h , and the Birkhoff sum S k f ( x ) is nonzero, then f does not satisfy Condition (H). 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