-odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1....
Transcript of -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1....
![Page 1: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/1.jpg)
α-odd continued fractions
Claire Merriman
January 5, 2020
The Ohio Sate University
![Page 2: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/2.jpg)
Table of contents
1. Regular continued fractions
2. Nakada α-continued fractions
3. Odd integer continued fractions
4. α-expansions with odd partial quotients, (Boca-M, 2019)
5. Natural extensions
1
![Page 3: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/3.jpg)
Regular continued fractions
![Page 4: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/4.jpg)
Standard algorithm
Let x > 0, then a0 is the largest integer less than x .
x = a0 +1
r1for 1 < r1. Let a1 be the largest integer less that r1.
x = a0 +1
a1 +1
r2
for 1 < r2. Continue this process,
x = a0 +1
a1 +1
a2 + . . .
= [a0; a1, a2, . . . ] .
2
![Page 5: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/5.jpg)
Facts and tricks
• This process terminates if and only if x is rational.
• n = n − 1 + 11 , so we require that the last digit is greater than 1.
• Then every real number has a unique continued fraction expansion.
• 1[a0;a1,a2,... ] =
1
a0 + [0; a1, a2, . . . ]= [0; a0, a1, a2, . . . ].
• 1[0;a1,a2,... ] = a1 +
1
a2 + [0; a3, . . . ]= [a0; a1, a2, . . . ].
3
![Page 6: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/6.jpg)
Regular continued fraction Gauss map
Define T : [0, 1)→ [0, 1) by
T (x) =
1x −
⌊1x
⌋if x 6= 0
0 if x = 0=
1x − k for x ∈
(1
k+1 ,1k
]0 if x = 0
.
Note that T ([0; a1, a2, a3, . . . ]) = [0; a2, a3, . . . ]
4
![Page 7: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/7.jpg)
Natural Extension of the regular Gauss map
Define T : [0, 1)2 → [0, 1)2 by
T (x , y) =
(
1x − k , 1
y+k
)for x ∈
(1
k+1 ,1k
](0, y) if x = 0
.
T (([0; a1, a2, . . . ], [0; b1, b2, . . . ])) = ([0; a2, a3, . . . ], [0; a1, b1, b2, . . . ]).
5
![Page 8: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/8.jpg)
Facts about the regular Gauss map
• The dynamical system ([0, 1),T ,B[0,1), µ) is ergodic for
dµ = 1log 2
dxx+1
• The dynamical system ([0, 1)2,T ,B[0,1)2 , ν) is ergodic for
dν = 1log 2
dxdy(1+xy)2
6
![Page 9: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/9.jpg)
Nearest integer continued fractions
In the Euclidean algorithm definition of continued fractions, we could
take the nearest integer instead of the floor.
TNICF :[− 1
2 ,12
)→[− 1
2 ,12
)by
x 7→ 1
|x |−⌊
1
|x |+
1
2
⌋=
1
|x |− k for |x | ∈
(1
k − 12
,1
k + 12
)
For example,1
2 +1
1 +1
3 +1
3
=1
3−1
4 +1
3
7
![Page 10: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/10.jpg)
Nearest integer continued fractions
In the Euclidean algorithm definition of continued fractions, we could
take the nearest integer instead of the floor.
TNICF :[− 1
2 ,12
)→[− 1
2 ,12
)by
x 7→ 1
|x |−⌊
1
|x |+
1
2
⌋=
1
|x |− k for |x | ∈
(1
k − 12
,1
k + 12
)
For example,1
2 +1
1 +1
3 +1
3
=1
3−1
4 +1
3
7
![Page 11: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/11.jpg)
Nakada α-continued fractions
![Page 12: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/12.jpg)
Nakada α-continued fractions
Nakada (1981) introduced the α continued fractions.
Define fα on [α− 1, α) to be
fα(x) =1
|x |−⌊
1
|x |+ 1− α
⌋=
1
|x |− k for |x | ∈
(1
k + α,
1
k − 1 + α
]Regular continued fractions when α = 1
Nearest integer continued fractions when α = 12
8
![Page 13: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/13.jpg)
Nakada α-Gauss map
'''' ' 'α1
α+2
1 2
1
α+3
1 3
α-1
1 2
-1
α+3
-1
α+2
1 3
-1
α+4
1 4
... ...
9
![Page 14: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/14.jpg)
Nakada α-continued fractions
Define fα on [α− 1, α) to be
fα(x) =1
|x |−⌊
1
|x |+ 1− α
⌋=
1
|x |− k for |x | ∈
(1
k + α,
1
k − 1 + α
]
Then the first digit of x is (sign(x)/k) εi = sign f i−1α (x), ai = k for
|f i−1α (x)| ∈
(1
k+α ,1
k−1+α
]
x =ε1
a1 +ε2
a2 +ε3
. . .
= [(ε1/a1)(ε2/a2), . . . ]α
10
![Page 15: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/15.jpg)
Natural Extension
Define fα : Ωα → Ωα by
(x , y) 7→(
1
|x |− a1,
1
a1 + εy
)for |x | ∈
(1
a1 − 1 + α,
1
a1 + α
)
ε0
a0 +ε1
a1 + . . .
,1
a−1 +ε−1
a−2 + . . .
7→ ε1
a1 +ε2
a2 + . . .
,1
a0 +ε0
a−1 + . . .
11
![Page 16: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/16.jpg)
Natural Extension
Define fα : Ωα → Ωα by
(x , y) 7→(
1
|x |− a1,
1
a1 + εy
)for |x | ∈
(1
a1 − 1 + α,
1
a1 + α
)
ε0
a0 +ε1
a1 + . . .
,1
a−1 +ε−1
a−2 + . . .
7→ ε1
a1 +ε2
a2 + . . .
,1
a0 +ε0
a−1 + . . .
11
![Page 17: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/17.jpg)
What is Ωα?
αα-1 f(α)
g
1
1-g
1/2
αα-1 f(α) f(α-1)
g
1-g
1/2
αα-1 f(α)f(α-1)
g
1-g
1/2
Figure 1: g < α ≤ 1 12≤ α ≤ g
√2− 1 ≤ α ≤ 1
2
The natural extension domain Ωα
12
![Page 18: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/18.jpg)
α <√
2− 1
α-1 fα(α-1) fα(α) αfα2(α-1)fα
2(α)
1+ 1
2
1- 5
1
2
5 -1
1
2
13
![Page 19: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/19.jpg)
Odd integer continued fractions
![Page 20: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/20.jpg)
Odd integer algorithm
What if instead of rounding down in the Euclidean algorithm, we found
the nearest odd integer?
1. How do we deal with even integers?
2. How do we get all positive real numbers?
14
![Page 21: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/21.jpg)
Odd integers, an example
Let x = 3613 .
• Since x ∈ (2, 3), x = 3− r1 for r1 ∈ (0, 1).
Here, r1 = 313 .
• x = 3−1133
= 3−1
5− r2for r2 ∈ (0, 1). Here, r2 = 2
3 .
• x = 3−1
5−132
= 3−1
5−1
1 + r3
for r3 ∈ (0, 1). Here, r3 = 12 .
• x = 3−1
5−1
1 +1
1 + 1
= 3−1
5−1
1 +1
3− 1
15
![Page 22: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/22.jpg)
Odd integers, an example
Let x = 3613 .
• Since x ∈ (2, 3), x = 3− r1 for r1 ∈ (0, 1). Here, r1 = 313 .
• x = 3−1133
= 3−1
5− r2for r2 ∈ (0, 1).
Here, r2 = 23 .
• x = 3−1
5−132
= 3−1
5−1
1 + r3
for r3 ∈ (0, 1). Here, r3 = 12 .
• x = 3−1
5−1
1 +1
1 + 1
= 3−1
5−1
1 +1
3− 1
15
![Page 23: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/23.jpg)
Odd integers, an example
Let x = 3613 .
• Since x ∈ (2, 3), x = 3− r1 for r1 ∈ (0, 1). Here, r1 = 313 .
• x = 3−1133
= 3−1
5− r2for r2 ∈ (0, 1). Here, r2 = 2
3 .
• x = 3−1
5−132
= 3−1
5−1
1 + r3
for r3 ∈ (0, 1).
Here, r3 = 12 .
• x = 3−1
5−1
1 +1
1 + 1
= 3−1
5−1
1 +1
3− 1
15
![Page 24: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/24.jpg)
Odd integers, an example
Let x = 3613 .
• Since x ∈ (2, 3), x = 3− r1 for r1 ∈ (0, 1). Here, r1 = 313 .
• x = 3−1133
= 3−1
5− r2for r2 ∈ (0, 1). Here, r2 = 2
3 .
• x = 3−1
5−132
= 3−1
5−1
1 + r3
for r3 ∈ (0, 1). Here, r3 = 12 .
• x = 3−1
5−1
1 +1
1 + 1
= 3−1
5−1
1 +1
3− 1
15
![Page 25: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/25.jpg)
Odd integers, an example
Let x = 3613 .
• Since x ∈ (2, 3), x = 3− r1 for r1 ∈ (0, 1). Here, r1 = 313 .
• x = 3−1133
= 3−1
5− r2for r2 ∈ (0, 1). Here, r2 = 2
3 .
• x = 3−1
5−132
= 3−1
5−1
1 + r3
for r3 ∈ (0, 1). Here, r3 = 12 .
• x = 3−1
5−1
1 +1
1 + 1
= 3−1
5−1
1 +1
3− 1
15
![Page 26: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/26.jpg)
Another example
G =1 +√
5
2= 1 +
1
1 +1
1 +1
1 + . . .
=1
1−1
3−1
3− . . .
16
![Page 27: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/27.jpg)
Odd Continued fractions
Rieger (1980) defined a new type of continued fractions
x = a0 +ε1
a1 +ε2
a2 +ε3
. . .
= a0 + 〈〈(ε1/a1)(ε2/a2), . . .〉〉o
where the ai are odd, εi = ±1, and ai + εi+1 ≥ 2. That is, you never
have a 1 followed by subtraction.
17
![Page 28: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/28.jpg)
The extended odd Gauss Map
Define To on [−1, 1)
To(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 2,
1
2k
]Then the first digit of x is (sign(x)/(2k + 1)).
εi = signT i−10 (x), ai = 2k + 1 for |T i−1
o (x)| ∈(
12k+2 ,
12k
]
18
![Page 29: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/29.jpg)
Odd Gauss Map
'''''' ' ' '1
1
2
(+1/1)
1
4
(+1/3)
1
6
(+1/5)
-1
(-1/1)
-1
4
-1
6
(-1/3) (-1/5)
... ...
To
(〈〈(ε1/a1)(ε2/a2), . . .〉〉o
)= 〈〈(ε2/a2)(ε3/a3), . . .〉〉o .
19
![Page 30: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/30.jpg)
Grotesque Continued Fractions
Rieger (1980) defined the grostesque continued fractions
y =ε0
a0 +ε−1
a−1 +ε−2
. . .
= 〈〈(ε1/a1)(ε2/a2), . . .〉〉grot
where the ai are odd, εi = ±1, and ai + εi ≥ 2.
• −11+z is only allowed in the odd continued fractions
• 11−z is only permitted in grotesque continued fractions
20
![Page 31: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/31.jpg)
Grotesque Gauss map
Define Tgrot : [G − 2,G )→ [G − 2,G ) where G = 1+√
52 by
Tgrot(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 1 + G,
1
2k − 1 + G
]Then the first digit of x is (sign(x)/(2k + 1))
εi = signT i−1grot (x), ai = 2k + 1 for |T i−1
grot (x)| ∈(
12k+1+G ,
12k−1+G
]
21
![Page 32: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/32.jpg)
Grotesque Gauss Map
'''''''' ' ' '
...
G=1
G-1
1
G+1
(+1/1)
1
G+3
(+1/3)(+1/5)
G-2=-1
G+1
(-1/3)
-1
G+3
(-1/5)
Tgrot
(〈〈(ε1/a1)(ε2/a2), . . .〉〉grot
)= 〈〈(ε2/a2)(ε3/a3), . . .〉〉grot .
22
![Page 33: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/33.jpg)
What are we doing?
The odd continued fractions and grotesque continued fractions are dual
continued fraction expansions.
Goals:
1. Define a way to interpolate between the grotesque continued
fractions and the (extended) odd continued fractions.
2. Find invariant measures and natural extensions of the relevant Gauss
maps.
3. Extend these results to other dynamical systems that generate
continued fractions with odd denominators.
23
![Page 34: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/34.jpg)
α-expansions with odd partial
quotients, (Boca-M, 2019)
![Page 35: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/35.jpg)
Comparing Gauss maps
If we ignore the rules for generating the digits, both To and Tgrot take
x 7→ 1
|x |− (2k + 1)
〈〈(ε1/a1)(ε2/a2) . . .〉〉 7→ 〈〈(ε2/a2)(ε3/a3) . . .〉〉
To : [−1, 1)→ [−1, 1) Tgrot : [G − 2,G )→ [G − 2,G )
Intervals |x | ∈(
12k+2 ,
12k
]Intervals |x | ∈
(1
2k+1+G ,1
2k−1+G
]
(εi/1) always followed by (+1/ai+1) Never (−1/1).
Define a new map ϕα : [α− 2, α)→ [α− 2, α) by
ϕα(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 1 + α,
1
2k − 1 + α
]
24
![Page 36: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/36.jpg)
Comparing Gauss maps
If we ignore the rules for generating the digits, both To and Tgrot take
x 7→ 1
|x |− (2k + 1)
〈〈(ε1/a1)(ε2/a2) . . .〉〉 7→ 〈〈(ε2/a2)(ε3/a3) . . .〉〉
To : [−1, 1)→ [−1, 1) Tgrot : [G − 2,G )→ [G − 2,G )
Intervals |x | ∈(
12k+2 ,
12k
]Intervals |x | ∈
(1
2k+1+G ,1
2k−1+G
](εi/1) always followed by (+1/ai+1) Never (−1/1).
Define a new map ϕα : [α− 2, α)→ [α− 2, α) by
ϕα(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 1 + α,
1
2k − 1 + α
]
24
![Page 37: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/37.jpg)
Comparing Gauss maps
If we ignore the rules for generating the digits, both To and Tgrot take
x 7→ 1
|x |− (2k + 1)
〈〈(ε1/a1)(ε2/a2) . . .〉〉 7→ 〈〈(ε2/a2)(ε3/a3) . . .〉〉
To : [−1, 1)→ [−1, 1) Tgrot : [G − 2,G )→ [G − 2,G )
Intervals |x | ∈(
12k+2 ,
12k
]Intervals |x | ∈
(1
2k+1+G ,1
2k−1+G
](εi/1) always followed by (+1/ai+1) Never (−1/1).
Define a new map ϕα : [α− 2, α)→ [α− 2, α) by
ϕα(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 1 + α,
1
2k − 1 + α
]
24
![Page 38: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/38.jpg)
α-odd Gauss map
We will consider α ∈ [g ,G ) for g = −1+√
52 = G − 1.
ϕα : [α− 2, α)→ [α− 2, α) by
ϕα(x) =1
|x |− (2k + 1) for |x | ∈
(1
2k + 1 + α,
1
2k − 1 + α
]
Again, we define εi = sign(ϕi−1α (x)), ai = 2k + 1 such that
|ϕi−1α (x)| ∈
(1
2k+1+α ,1
2k−1+α
]ϕα(〈〈(ε1/a1)(ε2/a2)(ε3/a3) . . .〉〉α) = 〈〈(ε2/a2)(ε3/a3) . . .〉〉α
25
![Page 39: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/39.jpg)
Digit restrictions
Let dα(x) = 2b 12|x| −
1−α2 c+ 1 = a1(x) emphasizing the α.
Lemma
1. When g < α ≤ G we have dα(α) = 1 and when g ≤ α < G we have
dα(α− 2) = 1.
2. When α ∈ (g , 1) ∪ (1,G ) we have
1
ϕα(α)+
1
ϕα(α− 2)= −2
26
![Page 40: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/40.jpg)
Digit restrictions
3. When g < α ≤ 1 we have
− 1
3 + α< ϕα(α− 2) ≤ 0 ≤ ϕα(α) <
1
1 + α,
dα(ϕα(α)) = dα(ϕα(α− 2))− 2
and ϕ2α(α) = ϕ2
α(α− 2).
(1)
4. When 1 ≤ α < G we have
− 1
1 + α< ϕα(α) ≤ 0 ≤ ϕα(α− 2) < α,
dα(ϕα(α)) = dα(ϕα(α− 2)) + 2
and ϕ2α(α) = ϕ2
α(α− 2).
(2)
27
![Page 41: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/41.jpg)
Natural extensions
![Page 42: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/42.jpg)
Natural extension of the α-odd Gauss map
We define ϕα : Ωα → Ωα by
ε1
a1 +ε2
a2 + . . .
,1
a0 +ε0
a−1 + . . .
7→
ε2
a2 +ε3
a3 + . . .
,1
a1 +ε1
a0 +ε0
a−1 + . . .
Or
(x , y) 7→(
1
|x |− (2k + 1),
1
2k + 1 + ε(x)y
)for |x | ∈
(1
2k + 1 + α,
1
2k − 1 + α
], ε(x) = sign(x).
28
![Page 43: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/43.jpg)
Natural extension of the grotesque Gauss map
We define T grot : [G − 2,G )× [0, 1)→ [G − 2,G )× [0, 1) by
ε1
a1 +ε2
a2 + . . .
,1
a0 +ε0
a−1 + . . .
7→
ε2
a2 +ε3
a3 + . . .
,1
a1 +ε1
a0 +ε0
a−1 + . . .
Or
(x , y) 7→(
1
|x |− (2k + 1),
1
2k + 1 + ε(x)y
)for |x | ∈
(1
2k + 1 + G,
1
2k − 1 + G
], ε(x) = sign(x).
29
![Page 44: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/44.jpg)
Natural extension of the extended odd Gauss map
We define T 0 :([−1, 0)× [0, 2− G )
)∪([0, 1)× [0,G )
)→(
[−1, 0)× [0, 2− G ))∪([0, 1)× [0,G )
)by
ε1
a1 +ε2
a2 + . . .
,1
a0 +ε0
a−1 + . . .
7→
ε2
a2 +ε3
a3 + . . .
,1
a1 +ε1
a0 +ε0
a−1 + . . .
Or
(x , y) 7→(
1
|x |− (2k + 1),
1
2k + 1 + ε(x)y
)for |x | ∈
(1
2k + 2,
1
2k
], ε(x) = sign(x).
30
![Page 45: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/45.jpg)
Natural extension domains
ϕα(α)0 1/(1+α)-1/(3+α)
2-G
1
0αα-2
G
ϕα(α-2) ϕα(α) 0-1/(1+α)
2-G
1
0αα-2
G
ϕα(α-2)
Figure 2: The natural extension domains Ωα for g ≤ α ≤ 1 (left) and
1 ≤ α ≤ G (right)
31
![Page 46: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/46.jpg)
Some ergodic theory results
(Boca-M, 2019)
• The measure .µα = 13 log G
dxdy(1+xy)2 is Φα-invariant on Ωα.
• dνα = hαdλ is ϕα-invariant with hα coming from integrating .µαover the y -values of the natural extension domain.
• (Ωα,BΩα, µα,Φα) gives the natural extension of (Iα,BIα , να, ϕα)
• (Iα,BIα , να, ϕα) is ergodic, exact, and mixing of all orders.
32
![Page 47: -odd continued fractionsnumeration/uploads/Main/merriman... · 2021. 1. 4. · Table of contents 1. Regular continued fractions 2. Nakada -continued fractions 3. Odd integer continued](https://reader035.fdocument.org/reader035/viewer/2022081411/60ab956730ab90439f432bfb/html5/thumbnails/47.jpg)
Standard consequences of ergodicity
Let pnqn
= 〈〈(ε1/a1) . . . (εn/an)〉〉α,
For every α ∈ [g ,G ] and almost every x ∈ Iα:
(i) limn
1
nlog |qn(x ;α)x − pn(x ;α)| = − π2
18 logG.
(ii) limn
1
nlog qn(x ;α) =
π2
18 logG.
(iii) limn
1
nlog
∣∣∣∣x − pn(x ;α)
qn(x ;α)
∣∣∣∣ = − π2
9 logG.
For every α ∈ [g ,G ] the entropy of ϕα with respect to να is
H(α) = −2
∫ α
α−2
log |x | hα(x) dx =π2
9 logG.
33