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Invariant measures for Moebius maps with three branches
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Invariant measures for Moebius maps with three branches Fritz Schweiger FB Mathematik, Universität Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria
a r t i c l e
i n f o
Article history: Received 11 February 2017 Received in revised form 2 August 2017 Accepted 2 August 2017 Available online xxxx Communicated by the Principal Editors
a b s t r a c t This note continues the investigations of invariant measures for so-called Moebius maps in [9] and [7]. Here two families of Moebius maps with three branches are constructed for which the invariant densities can be written down as an infinite series. In certain cases these densities are not rational functions. © 2017 Elsevier Inc. All rights reserved.
MSC: 11K55 28D05 37A05 Keywords: f-Expansion Invariant measures
1. Introduction In recent years the ergodic theory of non-invertible maps has been of interest. A possible frame work for these maps is the concept of fibred systems [6]. A map T : B → B is called a fibred systems if there is a partition {B(k) : k ∈ I} of B with the index set I that is finite or countable, such that the restriction of T to B(k) is injective. This restriction is called a branch of T . E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2017.08.019 0022-314X/© 2017 Elsevier Inc. All rights reserved.
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Many maps of this type can be found in number theory. We give two examples. First we mention T x = gx − ε, ε = ε(x) = gx, g ≥ 2 is an integer.
(1)
This map is connected with g-adic expansions x=
∞ εj . j g j=1
We also have the Gauss map Tx =
1 1 − a, a = a(x) = . x x
(2)
This map leads to regular continued fractions 1
x=
1
a1 + a2 +
.
1 a3 + ...
A short account on the historical development of the so-called f -expansions can be found in [8]. A lot of work has already done on the ergodicity and the existence of invariant measures for such maps. Note that the existence of a finite invariant measure can be used to calculate probabilities by the use of ergodic theorems (see [6]). More on infinite measures can be found in [1]. In many cases the existence of an invariant measure can proved by the verification of some sufficient conditions. Much less is known about the shape of an invariant density. Lebesgue measure is invariant for g-adic expansions. Due to an old conjecture of Gauss 1 the invariant density for continued fractions is known, namely h(x) = 1+x . One extension of continued fractions to higher dimensions is the Jacobi–Perron algorithm. For n = 2 the following map T (x, y) =
y 1 y 1 − a, − b), a = a(x) = , b = b(x) = x x x x
is ergodic and admits a finite invariant measure but the shape of the density is not known. Another tricky example is the Bolyai algorithm which is related to the map √ T : [1, 2] → [1, 2], T x = x2 , 1 ≤ x < 2; √ √ √ T x = x2 − 1, 2 ≤ x < 3; T x = x2 − 2, 3 ≤ x < 2. The shape of its invariant density is not known [3], [9].
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The density of the following map which is related to continued fraction with variable numerators is also unknown [2]. Tx =
a 1 − (a2 + j), a = a(x) = , j = j(x) = ax. x x
This note continues the investigations of [9] and [7]. In [7] the name Moebius maps has been proposed for fractional linear maps T : [0, 1] → [0, 1] with three branches. In this paper we use partition points 0, 13 , 23 , 1. The inverse branches are given by three matrices B labeled λ, μ, ν out of the following list 3 3λ − 3 3 3μ − 3 3 3ν − 3 , , 0 λ 1 2μ − 1 2 3ν − 2 3 λ−3 3 3μ − 3 1 3ν − 1 , , . 1 −1 2 μ−2 1 2ν − 1 The parameters satisfy some conditions to avoid attractive fixed points or poles inside the relevant interval of definition. We call the map T of type (ε1 , ε2 , ε3 ), ε = 1 or − 1 if the branch is increasing (ε = 1) or decreasing (ε = −1). An important idea is to try to find a dual Moebius map. This is a Moebius map on an interval [γ, δ] which is built up from the transposed matrices of the given fractional linear maps. It generalizes the notion of a backward continued fraction (see [5], [4]). Then the invariant density has the form δ h(x) =
dy (1 + xy)2
γ
[6], [8], [7]. One likely method to do this is to try to find an integral matrix A such that A × B = B # × A where B # denotes the transposed matrix of B. A second method looks for other dual maps which are not related in this way. These are called exceptional duals. Both methods allow us to calculate the interval [γ, δ] from the parameters λ, μ, ν. In this note we construct two families of maps with a quite different invariant density. Note that we do not consider normalization of the invariant densities. 2. The construction Starting with a suitable piecewise fractional linear map we construct two families of Moebius maps with three branches for which the invariant densities can be written down as an infinite series. In certain cases these densities are not rational functions. Theorem 1. In the following cases a Moebius map S with two branches is given and a Moebius map T with three branches can be constructed such that its invariant density can be written down as an infinite series.
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(a) W0 (x) =
λx 1 + 5x , W1 (x) = 3 + (3λ − 3)x 3 + 3x
W0 (x) =
λx 3−x , W1 (x) = 3 + (3λ − 3)x 3 + 3x
W0 (x) =
1−x 1 + 5x , W1 (x) = 3 + (λ − 3)x 3 + 3x
W0 (x) =
1−x 3−x , W1 (x) = . 3 + (λ − 3)x 3 + 3x
(b)
(c)
(d)
Proof. We start with a fractional linear map S : [0, 1] → [0, 1] with two branches and the partition points 0, 13 , and 1. The first branch contains the free parameter λ, 0 < λ ≤ 3. The second branch is chosen such that W1 ( 13 ) = 23 . It is easy to calculate the invariant densities for these cases: • (a) γ, δ ∈ { 3λ−3 3−λ , 3} 2−λ • (b) γ, δ ∈ { 3λ−3 3−λ , λ }
• (c) γ, δ ∈ { λ−6 6 , 3} • (d) γ, δ ∈ { 3λ−12 , λ−12 8 4−3λ } We form the jump transformation T : [0, 1] → [0, 1] defined by T x = Sx, 0 ≤ x <
1 1 , T x = S 2 x, ≤ x ≤ 1. 3 3
The device of jump transformations is explained in [6]. Then the three branches W0 , W10 = W1 ◦ W0 , and W11 = W1 ◦ W1 form a Moebius map. The type and the new parameters μ and ν are given by: • • • •
(a) (1, 1, 1), μ = 4λ 3 ,ν = 3 (b) (1, 1, −1), μ = 43 , ν = 4λ 9 (c) (−1, −1, 1), μ = λ4 , ν = 3 (d) (−1, 1, 1), μ = 43 , ν = λ4
If g(x) is the invariant density for T then the relation h(x) = g(x) + g(W1 x)ω1 (x)
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holds. Here as usual ω1 denotes the Jacobian of W1 . We calculate the density g = g(x) in the following way. Form the jump transformation R of the map S which avoids the set [ 13 , 1]. This amounts to Rx = S n+1 x if n is the first entry time of the point x to the set [0, 13 [. If κ = κ(x) is the invariant density for R then h(x) = κ(x) + κ(W1 x)ω1 (x) + κ(W11 x)ω11 (x) + κ(W111 x)ω111 (x) + ... . Since h(x) is a rational function almost all poles in this series will cancel. The relation κ(x) = h(W0 x)ω0 (x) allows one to calculate κ(x). We now form the jump transformation of T which avoids the interval with endpoints W11 (0) and W11 (1). Both jump transformations coincide and we get g(x) = κ(x) + κ(W11 x)ω11 (x) + κ(W1111 x)ω1111 (x) + ... . Generally, there are three cases. (I) The series for g(x) contains infinitely many poles in case (a) and (c). Hence, the density g(x) is not a rational function. In case (a) we find ∞ 3λ − 3 3 − = h(x) = (Gn+1 (x) − Gn (x)) 1 + 3x 3 − λ + (3λ − 3)x n=0
with Gn (x) =
λ(6n + 2n ) − 2n+1 . 2λ(6n−1 − 2n−1 ) + 2n+1 + x(λ(6n + 2n ) − 2n+1 )
In case (c) the density is given as h(x) =
∞ λ−6 3 − = (Gn+1 (x) − Gn (x)) 1 + 3x 6 + (λ − 6)x n=0
with Gn (x) =
λ(6n
−
2n )
λ(3 · 6n + 2n ) − 3 · 2n+3 . + 3 · 2n+3 + x(λ(3 · 6n + 2n ) − 3 · 2n+3 )
(II) The series for g(x) converges to the density given by an exceptional dual in case (b) and (d). √ In case (b) the relevant condition is 3λν + 3ν = λ2 + λμ. If λ < −1+2 13 then the √ branches of the dual algorithm are ordered as λνμ, if λ > −1+2 13 their order is μνλ.
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√
In case (d) the condition is λ = 3μν. If λ < 20+9 832 then the order is νλμ, if √ λ > 20+9 832 it is μλν. 1 (III) If γ = δ then up to normalization h(x) = g(x) = (1+γx) 2 . This case could be seen as a special case of (II) but it is the unique case when an integral matrix A such that A × B = B # × A exists. The condition γ = δ leads to • • • •
(a) λ = 2 (b) λ is the positive root of λ2 + λ − 3 = 0 (c) λ = 24 (d) λ is the positive root of 9λ2 − 40λ − 48 = 0
We first give an example for case (a). λ = 1, μ = h(x) =
4 ,ν=3 3
1 1 , κ(x) = . 1 + 3x 1+x
Then 1 3 9 1 1 + + + + ... = . 1 + x (1 + 2x)(1 + x) (5 + 13x)(1 + 2x) (7 + 20x)(5 + 13x) 1 + 3x But g(x) =
3 27 1 + + + ... . 1 + x (5 + 13x)(1 + 2x) (41 + 121x)(7 + 20x)
An example for case (b) is given by λ = 1, μ =
4 4 ,ν= 3 9
1 1 , κ(x) = 1+x 3+x 1 1 6 9 1 + + + + ... = . 3 + x (3 + 2x)(1 + x) (5 + 7x)(3 + x) (33 + 19x)(3 + 2x) 1+x h(x) =
However, g(x) =
6 27 η 1 + + + ... = 3 + x (15 + 7x)(3 + x) (39 + 20x)(15 + 7x) 1 + ηx
where η is the positive root of 3η 2 + 4η − 3 = 0. The exceptional dual is given by the map T # : [0, η] → [0, η] and the matrices
3 0
0 1
,
9 9 3 −1
,
9 3 3 5
.
2
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Theorem 2. In the following cases a Moebius map S with two branches is given and a Moebius T with three branches can be constructed such that its invariant density can be written down as an infinite series. (a) W0 (x) =
2x 2 + (3ν − 2)x , W1 (x) = 6 − 3x 3 + (3ν − 3)x
W0 (x) =
2x 1 + (2ν − 1)x , W1 (x) = 6 − 3x 1 + (3ν − 1)x
W0 (x) =
4 − 4x 2 + (3ν − 2)x , W1 (x) = 6 − 3x 3 + (3ν − 3)x
W0 (x) =
4 − 4x 1 + (2ν − 1)x , W1 (x) = . 6 − 3x 1 + (3ν − 1)x
(b)
(c)
(d)
Proof. As before we start with a fractional linear map S : [0, 1] → [0, 1] with two branches and the partition points 0, 23 , and 1. The first branch is chosen such that W0 ( 23 ) = 13 . The second branch contains the free parameter ν. It is easy to calculate the invariant densities in these cases. They are given by: • • • •
(a) γ, δ ∈ {− 34 , −3+3ν } 2 (b) γ, δ ∈ {− 34 , 6ν − 1} 3ν−3 (c) γ, δ ∈ { 1−2ν 2ν , 2 } 3−12ν (d) γ, δ ∈ { 1−12ν 8ν+2 , 4ν−3 }
We form the jump transformation T : [0, 1] → [0, 1] defined by T x = Sx,
2 2 ≤ x < 1, T x = S 2 x, 0 ≤ x ≤ . 3 3
Then the three branches W1 , W01 = W0 ◦ W1 , and W00 = W0 ◦ W0 form a Moebius map. The type and the new parameters λ and μ are given below. • • • •
(a) (1, 1, 1), λ = 13 , μ = 3ν 4 (b) (1, −1, −1), λ = 13 , μ = 4ν 3 (c) (−1, 1, 1), λ = 9ν 4 ,μ = 4 (d) (1, 1, −1), λ = 4ν, μ = 34
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If g(x) is the invariant density for T then the relation h(x) = g(x) + g(W0 x)ω0 (x) holds. We calculate the density g = g(x) in the following way. We form the jump transformation R of the map S which avoids the set [0, 23 ]. This amounts to Rx = S n+1 x if n is the first entry time of the point x to the set [ 23 , 1[. If κ = κ(x) is the invariant density for R then h(x) = κ(x) + κ(W0 x)ω0 (x) + κ(W00 x)ω00 (x) + κ(W000 x)ω000 (x) + ... . Again, since h(x) is a rational function almost all poles in this series will cancel. The relation κ(x) = h(W1 x)ω1 (x) allows to calculate κ(x). We form the jump transformation of T which avoids the interval with endpoints W00 (0) and W00 (1). Both jump transformations coincide and we get g(x) = κ(x) + κ(W00 x)ω00 (x) + κ(W0000 x)ω0000 (x) + ... . Generally, there are three cases. (I) The series for g(x) contains infinitely many poles in case (a) and (b). Hence, the density g(x) is not a rational function. As before we find for case (a) h(x) =
∞ 3 3ν − 3 + = (Gn+1 (x) − Gn (x)) 2 + (3ν − 3)x 4 − 3x n=0
with Gn (x) =
6n − 2n (2ν − 1) 8 · 6n−1 + x(2n (2ν − 1) − 6n )
and for case (b) h(x) =
∞ 3 6ν − 1 + = (Gn+1 (x) − Gn (x)) 1 + (6ν − 1)x 4 − 3x n=0
with Gn (x) =
6n
2n−2 (3n+1 + 1 − 24ν) . + x(2n−2 (24ν − 3n+1 − 1))
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(II) The series for g(x) converges to the density given by an exceptional dual in case (c) and (d). √ For case (d) the relevant condition is 3λν + 3ν = λ2 + λμ. If ν < −20+48 831 then the √ order is λνμ, if ν > −20+48 831 it is μνλ. √ The condition for case (c) it is λ = 3μν. If ν < 1+6 13 the order in the exceptional √ dual is νλμ, if ν > 1+6 13 it is μλν. 1 (III) If γ = δ then up to normalization h(x) = g(x) = (1+γx) 2 . This case could be seen as a special case of (II) but it is the unique case when an integral matrix A such that A × B = B # × A exists. The condition γ = δ leads to • • • •
(a) ν = 12 1 (b) ν = 24 (c) ν is the positive root of 3ν 2 − ν − 1 = 0 (d) ν is the positive root of 48ν 2 + 40ν − 9 = 0.
We give an example for case (c): ν = 1, λ =
3 9 ,μ= 4 4
1 1 , κ(x) = 2−x 4−x 1 6 9 1 1 + + + = . 4 − x (5 − 2x)(2 − x) (22 − 7x)(4 − x) (52 − 19x)(5 − 2x) 1 − 2x h(x) =
But g(x) =
1 6 27 η = + + + ... , 1 − ηx 4 − x (22 − 7x)(4 − x) (59 − 20x)(22 − 7x)
where η is the positive root of 4η 2 − 10η + 3 = 0 which belongs to an exceptional dual on the interval [−η, 0]. 2 References [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, American Mathematical Society, 1997. [2] Karma Dajani, Cor Kraaikamp, Niels Daniël Simon Langeveld, Continued fraction expansions with variable numerators, Ramanujan J. 37 (3) (2015) 617–639. [3] O. Jenkinson, M. Pollicott, Ergodic properties of the Bolyai–Rényi expansion, Indag. Math. (N.S.) 11 (2000) 399–418. [4] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981) 399–426. [5] H. Nakada, S. Ito, S. Tanaka, On the invariant measure for the transformations associated with some real continued fractions, Keio Eng. Rep. 30 (1977) 159–175. [6] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. [7] F. Schweiger, Differentiable equivalence of fractional linear maps, in: Dynamics & Stochastics, in: IMS Lecture Notes Monogr. Ser., vol. 48, 2006, pp. 237–247.
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[8] F. Schweiger, Continued Fractions and Their Generalizations: A Short History of f -Expansions Boston, Docent Press, Massachusetts, 2016. [9] F. Schweiger, Invariant measures of piecewise fractional linear maps and piecewise quadratic maps, Int. J. Number Theory (2017), forthcoming.
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Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
Invariant measures for Moebius maps with three branches Fritz Schweiger FB Mathematik, Universität Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria
a r t i c l e
i n f o
Article history: Received 11 February 2017 Received in revised form 2 August 2017 Accepted 2 August 2017 Available online xxxx Communicated by the Principal Editors
a b s t r a c t This note continues the investigations of invariant measures for so-called Moebius maps in [9] and [7]. Here two families of Moebius maps with three branches are constructed for which the invariant densities can be written down as an infinite series. In certain cases these densities are not rational functions. © 2017 Elsevier Inc. All rights reserved.
MSC: 11K55 28D05 37A05 Keywords: f-Expansion Invariant measures
1. Introduction In recent years the ergodic theory of non-invertible maps has been of interest. A possible frame work for these maps is the concept of fibred systems [6]. A map T : B → B is called a fibred systems if there is a partition {B(k) : k ∈ I} of B with the index set I that is finite or countable, such that the restriction of T to B(k) is injective. This restriction is called a branch of T . E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2017.08.019 0022-314X/© 2017 Elsevier Inc. All rights reserved.
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Many maps of this type can be found in number theory. We give two examples. First we mention T x = gx − ε, ε = ε(x) = gx, g ≥ 2 is an integer.
(1)
This map is connected with g-adic expansions x=
∞ εj . j g j=1
We also have the Gauss map Tx =
1 1 − a, a = a(x) = . x x
(2)
This map leads to regular continued fractions 1
x=
1
a1 + a2 +
.
1 a3 + ...
A short account on the historical development of the so-called f -expansions can be found in [8]. A lot of work has already done on the ergodicity and the existence of invariant measures for such maps. Note that the existence of a finite invariant measure can be used to calculate probabilities by the use of ergodic theorems (see [6]). More on infinite measures can be found in [1]. In many cases the existence of an invariant measure can proved by the verification of some sufficient conditions. Much less is known about the shape of an invariant density. Lebesgue measure is invariant for g-adic expansions. Due to an old conjecture of Gauss 1 the invariant density for continued fractions is known, namely h(x) = 1+x . One extension of continued fractions to higher dimensions is the Jacobi–Perron algorithm. For n = 2 the following map T (x, y) =
y 1 y 1 − a, − b), a = a(x) = , b = b(x) = x x x x
is ergodic and admits a finite invariant measure but the shape of the density is not known. Another tricky example is the Bolyai algorithm which is related to the map √ T : [1, 2] → [1, 2], T x = x2 , 1 ≤ x < 2; √ √ √ T x = x2 − 1, 2 ≤ x < 3; T x = x2 − 2, 3 ≤ x < 2. The shape of its invariant density is not known [3], [9].
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The density of the following map which is related to continued fraction with variable numerators is also unknown [2]. Tx =
a 1 − (a2 + j), a = a(x) = , j = j(x) = ax. x x
This note continues the investigations of [9] and [7]. In [7] the name Moebius maps has been proposed for fractional linear maps T : [0, 1] → [0, 1] with three branches. In this paper we use partition points 0, 13 , 23 , 1. The inverse branches are given by three matrices B labeled λ, μ, ν out of the following list 3 3λ − 3 3 3μ − 3 3 3ν − 3 , , 0 λ 1 2μ − 1 2 3ν − 2 3 λ−3 3 3μ − 3 1 3ν − 1 , , . 1 −1 2 μ−2 1 2ν − 1 The parameters satisfy some conditions to avoid attractive fixed points or poles inside the relevant interval of definition. We call the map T of type (ε1 , ε2 , ε3 ), ε = 1 or − 1 if the branch is increasing (ε = 1) or decreasing (ε = −1). An important idea is to try to find a dual Moebius map. This is a Moebius map on an interval [γ, δ] which is built up from the transposed matrices of the given fractional linear maps. It generalizes the notion of a backward continued fraction (see [5], [4]). Then the invariant density has the form δ h(x) =
dy (1 + xy)2
γ
[6], [8], [7]. One likely method to do this is to try to find an integral matrix A such that A × B = B # × A where B # denotes the transposed matrix of B. A second method looks for other dual maps which are not related in this way. These are called exceptional duals. Both methods allow us to calculate the interval [γ, δ] from the parameters λ, μ, ν. In this note we construct two families of maps with a quite different invariant density. Note that we do not consider normalization of the invariant densities. 2. The construction Starting with a suitable piecewise fractional linear map we construct two families of Moebius maps with three branches for which the invariant densities can be written down as an infinite series. In certain cases these densities are not rational functions. Theorem 1. In the following cases a Moebius map S with two branches is given and a Moebius map T with three branches can be constructed such that its invariant density can be written down as an infinite series.
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(a) W0 (x) =
λx 1 + 5x , W1 (x) = 3 + (3λ − 3)x 3 + 3x
W0 (x) =
λx 3−x , W1 (x) = 3 + (3λ − 3)x 3 + 3x
W0 (x) =
1−x 1 + 5x , W1 (x) = 3 + (λ − 3)x 3 + 3x
W0 (x) =
1−x 3−x , W1 (x) = . 3 + (λ − 3)x 3 + 3x
(b)
(c)
(d)
Proof. We start with a fractional linear map S : [0, 1] → [0, 1] with two branches and the partition points 0, 13 , and 1. The first branch contains the free parameter λ, 0 < λ ≤ 3. The second branch is chosen such that W1 ( 13 ) = 23 . It is easy to calculate the invariant densities for these cases: • (a) γ, δ ∈ { 3λ−3 3−λ , 3} 2−λ • (b) γ, δ ∈ { 3λ−3 3−λ , λ }
• (c) γ, δ ∈ { λ−6 6 , 3} • (d) γ, δ ∈ { 3λ−12 , λ−12 8 4−3λ } We form the jump transformation T : [0, 1] → [0, 1] defined by T x = Sx, 0 ≤ x <
1 1 , T x = S 2 x, ≤ x ≤ 1. 3 3
The device of jump transformations is explained in [6]. Then the three branches W0 , W10 = W1 ◦ W0 , and W11 = W1 ◦ W1 form a Moebius map. The type and the new parameters μ and ν are given by: • • • •
(a) (1, 1, 1), μ = 4λ 3 ,ν = 3 (b) (1, 1, −1), μ = 43 , ν = 4λ 9 (c) (−1, −1, 1), μ = λ4 , ν = 3 (d) (−1, 1, 1), μ = 43 , ν = λ4
If g(x) is the invariant density for T then the relation h(x) = g(x) + g(W1 x)ω1 (x)
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holds. Here as usual ω1 denotes the Jacobian of W1 . We calculate the density g = g(x) in the following way. Form the jump transformation R of the map S which avoids the set [ 13 , 1]. This amounts to Rx = S n+1 x if n is the first entry time of the point x to the set [0, 13 [. If κ = κ(x) is the invariant density for R then h(x) = κ(x) + κ(W1 x)ω1 (x) + κ(W11 x)ω11 (x) + κ(W111 x)ω111 (x) + ... . Since h(x) is a rational function almost all poles in this series will cancel. The relation κ(x) = h(W0 x)ω0 (x) allows one to calculate κ(x). We now form the jump transformation of T which avoids the interval with endpoints W11 (0) and W11 (1). Both jump transformations coincide and we get g(x) = κ(x) + κ(W11 x)ω11 (x) + κ(W1111 x)ω1111 (x) + ... . Generally, there are three cases. (I) The series for g(x) contains infinitely many poles in case (a) and (c). Hence, the density g(x) is not a rational function. In case (a) we find ∞ 3λ − 3 3 − = h(x) = (Gn+1 (x) − Gn (x)) 1 + 3x 3 − λ + (3λ − 3)x n=0
with Gn (x) =
λ(6n + 2n ) − 2n+1 . 2λ(6n−1 − 2n−1 ) + 2n+1 + x(λ(6n + 2n ) − 2n+1 )
In case (c) the density is given as h(x) =
∞ λ−6 3 − = (Gn+1 (x) − Gn (x)) 1 + 3x 6 + (λ − 6)x n=0
with Gn (x) =
λ(6n
−
2n )
λ(3 · 6n + 2n ) − 3 · 2n+3 . + 3 · 2n+3 + x(λ(3 · 6n + 2n ) − 3 · 2n+3 )
(II) The series for g(x) converges to the density given by an exceptional dual in case (b) and (d). √ In case (b) the relevant condition is 3λν + 3ν = λ2 + λμ. If λ < −1+2 13 then the √ branches of the dual algorithm are ordered as λνμ, if λ > −1+2 13 their order is μνλ.
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√
In case (d) the condition is λ = 3μν. If λ < 20+9 832 then the order is νλμ, if √ λ > 20+9 832 it is μλν. 1 (III) If γ = δ then up to normalization h(x) = g(x) = (1+γx) 2 . This case could be seen as a special case of (II) but it is the unique case when an integral matrix A such that A × B = B # × A exists. The condition γ = δ leads to • • • •
(a) λ = 2 (b) λ is the positive root of λ2 + λ − 3 = 0 (c) λ = 24 (d) λ is the positive root of 9λ2 − 40λ − 48 = 0
We first give an example for case (a). λ = 1, μ = h(x) =
4 ,ν=3 3
1 1 , κ(x) = . 1 + 3x 1+x
Then 1 3 9 1 1 + + + + ... = . 1 + x (1 + 2x)(1 + x) (5 + 13x)(1 + 2x) (7 + 20x)(5 + 13x) 1 + 3x But g(x) =
3 27 1 + + + ... . 1 + x (5 + 13x)(1 + 2x) (41 + 121x)(7 + 20x)
An example for case (b) is given by λ = 1, μ =
4 4 ,ν= 3 9
1 1 , κ(x) = 1+x 3+x 1 1 6 9 1 + + + + ... = . 3 + x (3 + 2x)(1 + x) (5 + 7x)(3 + x) (33 + 19x)(3 + 2x) 1+x h(x) =
However, g(x) =
6 27 η 1 + + + ... = 3 + x (15 + 7x)(3 + x) (39 + 20x)(15 + 7x) 1 + ηx
where η is the positive root of 3η 2 + 4η − 3 = 0. The exceptional dual is given by the map T # : [0, η] → [0, η] and the matrices
3 0
0 1
,
9 9 3 −1
,
9 3 3 5
.
2
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Theorem 2. In the following cases a Moebius map S with two branches is given and a Moebius T with three branches can be constructed such that its invariant density can be written down as an infinite series. (a) W0 (x) =
2x 2 + (3ν − 2)x , W1 (x) = 6 − 3x 3 + (3ν − 3)x
W0 (x) =
2x 1 + (2ν − 1)x , W1 (x) = 6 − 3x 1 + (3ν − 1)x
W0 (x) =
4 − 4x 2 + (3ν − 2)x , W1 (x) = 6 − 3x 3 + (3ν − 3)x
W0 (x) =
4 − 4x 1 + (2ν − 1)x , W1 (x) = . 6 − 3x 1 + (3ν − 1)x
(b)
(c)
(d)
Proof. As before we start with a fractional linear map S : [0, 1] → [0, 1] with two branches and the partition points 0, 23 , and 1. The first branch is chosen such that W0 ( 23 ) = 13 . The second branch contains the free parameter ν. It is easy to calculate the invariant densities in these cases. They are given by: • • • •
(a) γ, δ ∈ {− 34 , −3+3ν } 2 (b) γ, δ ∈ {− 34 , 6ν − 1} 3ν−3 (c) γ, δ ∈ { 1−2ν 2ν , 2 } 3−12ν (d) γ, δ ∈ { 1−12ν 8ν+2 , 4ν−3 }
We form the jump transformation T : [0, 1] → [0, 1] defined by T x = Sx,
2 2 ≤ x < 1, T x = S 2 x, 0 ≤ x ≤ . 3 3
Then the three branches W1 , W01 = W0 ◦ W1 , and W00 = W0 ◦ W0 form a Moebius map. The type and the new parameters λ and μ are given below. • • • •
(a) (1, 1, 1), λ = 13 , μ = 3ν 4 (b) (1, −1, −1), λ = 13 , μ = 4ν 3 (c) (−1, 1, 1), λ = 9ν 4 ,μ = 4 (d) (1, 1, −1), λ = 4ν, μ = 34
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If g(x) is the invariant density for T then the relation h(x) = g(x) + g(W0 x)ω0 (x) holds. We calculate the density g = g(x) in the following way. We form the jump transformation R of the map S which avoids the set [0, 23 ]. This amounts to Rx = S n+1 x if n is the first entry time of the point x to the set [ 23 , 1[. If κ = κ(x) is the invariant density for R then h(x) = κ(x) + κ(W0 x)ω0 (x) + κ(W00 x)ω00 (x) + κ(W000 x)ω000 (x) + ... . Again, since h(x) is a rational function almost all poles in this series will cancel. The relation κ(x) = h(W1 x)ω1 (x) allows to calculate κ(x). We form the jump transformation of T which avoids the interval with endpoints W00 (0) and W00 (1). Both jump transformations coincide and we get g(x) = κ(x) + κ(W00 x)ω00 (x) + κ(W0000 x)ω0000 (x) + ... . Generally, there are three cases. (I) The series for g(x) contains infinitely many poles in case (a) and (b). Hence, the density g(x) is not a rational function. As before we find for case (a) h(x) =
∞ 3 3ν − 3 + = (Gn+1 (x) − Gn (x)) 2 + (3ν − 3)x 4 − 3x n=0
with Gn (x) =
6n − 2n (2ν − 1) 8 · 6n−1 + x(2n (2ν − 1) − 6n )
and for case (b) h(x) =
∞ 3 6ν − 1 + = (Gn+1 (x) − Gn (x)) 1 + (6ν − 1)x 4 − 3x n=0
with Gn (x) =
6n
2n−2 (3n+1 + 1 − 24ν) . + x(2n−2 (24ν − 3n+1 − 1))
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(II) The series for g(x) converges to the density given by an exceptional dual in case (c) and (d). √ For case (d) the relevant condition is 3λν + 3ν = λ2 + λμ. If ν < −20+48 831 then the √ order is λνμ, if ν > −20+48 831 it is μνλ. √ The condition for case (c) it is λ = 3μν. If ν < 1+6 13 the order in the exceptional √ dual is νλμ, if ν > 1+6 13 it is μλν. 1 (III) If γ = δ then up to normalization h(x) = g(x) = (1+γx) 2 . This case could be seen as a special case of (II) but it is the unique case when an integral matrix A such that A × B = B # × A exists. The condition γ = δ leads to • • • •
(a) ν = 12 1 (b) ν = 24 (c) ν is the positive root of 3ν 2 − ν − 1 = 0 (d) ν is the positive root of 48ν 2 + 40ν − 9 = 0.
We give an example for case (c): ν = 1, λ =
3 9 ,μ= 4 4
1 1 , κ(x) = 2−x 4−x 1 6 9 1 1 + + + = . 4 − x (5 − 2x)(2 − x) (22 − 7x)(4 − x) (52 − 19x)(5 − 2x) 1 − 2x h(x) =
But g(x) =
1 6 27 η = + + + ... , 1 − ηx 4 − x (22 − 7x)(4 − x) (59 − 20x)(22 − 7x)
where η is the positive root of 4η 2 − 10η + 3 = 0 which belongs to an exceptional dual on the interval [−η, 0]. 2 References [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, American Mathematical Society, 1997. [2] Karma Dajani, Cor Kraaikamp, Niels Daniël Simon Langeveld, Continued fraction expansions with variable numerators, Ramanujan J. 37 (3) (2015) 617–639. [3] O. Jenkinson, M. Pollicott, Ergodic properties of the Bolyai–Rényi expansion, Indag. Math. (N.S.) 11 (2000) 399–418. [4] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981) 399–426. [5] H. Nakada, S. Ito, S. Tanaka, On the invariant measure for the transformations associated with some real continued fractions, Keio Eng. Rep. 30 (1977) 159–175. [6] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. [7] F. Schweiger, Differentiable equivalence of fractional linear maps, in: Dynamics & Stochastics, in: IMS Lecture Notes Monogr. Ser., vol. 48, 2006, pp. 237–247.
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[8] F. Schweiger, Continued Fractions and Their Generalizations: A Short History of f -Expansions Boston, Docent Press, Massachusetts, 2016. [9] F. Schweiger, Invariant measures of piecewise fractional linear maps and piecewise quadratic maps, Int. J. Number Theory (2017), forthcoming.