sup(x,y)∈X|det DΦ∆k(x, y)|
inf(x,y)∈X|det DΦ∆k(x, y)| =
à q3(k) 2q1(k)+q(k)3
!3
=
Ã4q1(k−1) + 3q(k−1)3 2q1(k−1)+q3(k−1)
!3
<
Ã
2 + q3(k−1) 2q1(k−1)+q3(k−1)
!3
< 33. Then we complete this lemma.
9.2 Weak Bernoulli properties of the modified negative slope al-
Lemma 9.8. (C.1) For any (x, y)6= (x0, y0) ∈X, there exists n≥0 such that Sn(x, y) and Sn(x0, y0) are not the same element in a partition of X.
Proof. It is easy to see that Ã
p(k)3 q(k)3 , r(k)3
q3(k)
! ,
Ã
p(k)1 +p(k)3
q(k)1 +q3(k), r1(k)+r(k)3 q1(k)+q(k)3
! ,
Ã
p(k)2 +p(k)3
q2(k)+q(k)3 , r(k)2 +r3(k) q(k)2 +q3(k)
! ,
and Ã
p(k)1 +p(k)2 +p(k)3
q1(k)+q2(k)+q3(k), r1(k)+r(k)2 +r3(k) q1(k)+q(k)2 +q3(k)
!
are Φ∆k(0,0), Φ∆k(1,0), Φ∆k(0,1), and Φ∆k(1,1), respectively. Then we show that the diameter ofh(ε1, n1, m1), (ε2, n2, m2), . . . , (εk, nk, mk)iis bounded by the distance between Φ∆k(0,1) and Φ∆k(1,0) as follows.
Letl be the line that passes Φ∆k(0,1) and Φ∆k(1,0), then we see that l : (q(k)1 +q3(k))(x+y)−((p(k)1 +p(k)3 ) + (r(k)1 +r3(k))) = 0.
Letd(k, x, y) be the distance between Φ∆k(0,1) and Φ∆k(1,0), h1(k, x, y) and h2(k, x, y) be the distances between l and Φ∆k(0,1), Φ∆k(1,0), respectively. Then we have
d(k, x, y) = vu ut
Ãp(k)1 +p(k)3
q1(k)+q(k)3 − p(k)2 +p(k)3 q2(k)+q(k)3
!2 +
Ãr(k)1 +r3(k)
q(k)1 +q3(k) − r(k)2 +r(k)3 q(k)2 +q3(k)
!2 ,
h1(k, x, y) =
¯¯
¯¯
¯(q1(k)+q3(k))p(k)3 +r(k)3
q(k)3 −(p(k)1 +p(k)3 +r1(k)+r(k)3 )
¯¯
¯¯
¯
√2(q(k)1 +q3(k)) ,
h1(k, x, y) =
¯¯
¯¯
¯(q1(k)+q3(k))p(k)1 +p(k)2 +p(k)3 +r(k)1 +r2(k)+r(k)3
q(k)1 +q2(k)+q3(k) −(p(k)1 +p(k)3 +r1(k)+r3(k))
¯¯
¯¯
¯
√2(q1(k)+q(k)3 ) . From Lemma 9.2 and (9.3), (9.4) and (9.5), we obtain
d(k, x, y) =
√2 q1(k)+q3(k), h1(k, x, y) = 1
√2q3(k)(q1(k)+q(k)3 ),
h1(k, x, y) = 1
√2(q1(k)+q(k)3 )(q1(k)+q2(k)+q(k)3 ).
These imply that the diameter of h(ε1, n1, m1), (ε2, n2, m2), . . . , (εk, nk, mk)i is bounded by d(k, x, y). In the following, we show that d(k, x, y) is monotone decreasing. Then we complete this lemma.
(i) If q(k−1)1 >0, then by Lemma 9.2, we see that q1(k)+q(k)3 =
(nk+mk−1)q(k−1)1 + (nk+mk)q3(k−1) if εk= +1 (nk+mk+ 1)q(k−1)1 + (nk+mk)q3(k−1) if εk =−1
> q1(k−1)+q3(k−1) for εk =±1.
(ii) If q(k−1)1 <0, then by Lemma 9.2, we see that q1(k)+q(k)3
=
(nk+mk−1)(q1(k−1) +q3(k−1)) +q(k−1)3 if εk= +1 (nk+mk−1)(q1(k−1) +q3(k−1)) + (2q1(k−1)+q3(k−1)) if εk =−1
> q(k−1)1 +q(k−1)3 for εk =±1.
This is the assertion of this lemma.
Lemma 9.9. (C.4) We have
X∞ k=1
λ(Dk) < ∞ where λ denotes the 2-dimensional Lebesgue measure.
Proof. It is easy to see that
h(−1,1,1), . . . , (−1,1,1
| {z }
k times
)i
= {(x, y)|2−k+ 1
k x≤y <1, k
k+ 1 − k
k+ 1x≤y <1}.
From Lemma 4.5, we obtain
λ(Dk) = 2
(k+ 1)(2k+ 1). This is the assertion of this lemma.
Then we obtain the following theorem by [15].
Theorem 9.10. There exists an absolutely continuous invariant probability measure η for S and (S, η) is exact.
Proof. We see that the modified negative slope algorithm satisfies (C.1) - (C.4) of Yuri’s conditions. Hence we complete the proof of Theorem 9.10 by [15].
Remark 9.11. The exactness implies not only ergodicity but also mixing of all degrees. In [4], they showed the explicit form of the density function dλdη, which we will see in §10, and its ergodicity.
Next we show the following theorem.
Theorem 9.12. (Rohlin’s formula) The entropy Hη(S) of (X, S, η) is given by Hη(S) =
Z
X
log |det DS|dη.
In the following, we show (C.5)–(C.8) of Yuri’s conditions, which imply this theorem.
Lemma 9.13. (C.5) Wk =
X∞ l=0
X
∆l∈Dl
Ã
sup
(x,y)∈(∪kj=1Bj)
|det DΦ∆l(x, y)|
!
<∞.
Proof. It is easy to see that
det DΦ∆l(x, y) = 1
(−lx−ly+ 2l+ 1)3
for ∆l =h(−1,1,1), . . . , (−1,1,1)i. Then we complete this lemma from Lemma 4.7.
Lemma 9.14. (C.6)
]D1 = 2.
Proof. This is obvious.
Lemma 9.15. (C.7) We have
sup(x,y)∈X|det DΦ∆k(x, y)|
inf(x,y)∈X|det DΦ∆k(x, y)| = O(k3) for ∆k = {h(+1,1,1), . . . , (+1,1,1
| {z }
k times
)i, h(−1,1,1), . . . , (−1,1,1
| {z }
k times
)i}.
Proof. These follow from Lemma 4.9 and Lemma 9.13.
Lemma 9.16. (C.8) The function log|det DS| is integrable with respect to λ.
Proof. We can complete this lemma by Lemma 4.10.
Then we finish the proof of the Theorem 9.12 by [15].
In the following, we show that the modified negative slope algorithm is weak Bernoulli.
Theorem 9.17. The modified negative slope algorithm with the absolutely continuous in- variant probability measure η is weak Bernoulli.
To prove this theorem, we show (C.4)∗ and (C.9) of Yuri’s conditions.
Lemma 9.18. (C.4)∗
X∞ k=1
λ(Dk)·logk < ∞.
Proof. Since we haveλ(Dk) = (k+1)(2k+1)2 from the proof of Lemma 9.9. This is the assertion of this lemma.
Lemma 9.19. (C.9) If h(ε1, n1, m1), (ε2, n2, m2), . . . , (εk, nk, mk)i ∈ Dkc
and h(ε2, n2, m2), . . . , (εk, nk, mk)i ∈ Dk−1, then we have h(ε1, n1, m1)i ∈ B1, that is, (ε1, n1, m1)6= (±1,1,1).
Proof. It is easy to see from the definitions of Dk and Bk.
SinceS satisfies (C.1)–(C.9) with (C.4)∗, it implies the assertion of Theorem 9.17 by [15].
10 Absolutely continuous invariant measure of the mod- ified negative slope algorithm
In [4], the density function of the absolutely continuous invariant probability measure was given as follows.
dη
dλ = 1
4 log 2
1
(x+y)(2−x−y). We see this formula by checking Kuzmin’s equation
f(x, y) = X
ε=±1,n,m≥1
f(Φ(ε,n,m)(x, y))|det Φ(ε,n,m)(x, y)|
where f(x, y) = (x+y)(2−x−y)1 .
In this section, we give the same result by a different way, that is called a “natural extension method”. This method was originally started by [10] for a class of continued fraction algorithms. Let X = X× {(−∞, 0)2∪(1,∞)2}. For (x, y, z, w) ∈X, we define a map S onX by
S(x, y, z, w)
=
³
n0(x, y)− (x+y)−1y , m0(x, y)−(x+y)−1x , n0(x, y)− (z+w)−1w , m0(x, y)−(z+w)−1z
´
if x+y > 1
³ 1−y
1−(x+y) −n(x, y), 1−(x+y)1−x −m(x, y), 1−(z+w)1−w −n(x, y), 1−(z+w)1−z −m(x, y)
´
if x+y < 1, where n0(x, y) = n(x, y) + 1 and m0(x, y) = m(x, y) + 1. Then it is easy to see that S is bijective on X except for the set of 4-dimensional Lebesgue measure 0.
Proposition 10.1. The measure η defined by dη
dλ = 1
|(x+y)−(z+w)|3
is an invariant measure for S, where λ denotes the 4-dimensional Lebesgue measure.
Proof. We complete this proposition by Proposition 5.1.
Corollary 10.2. The measureη defined by dη
dλ = 1
4 log 2
1
(x+y)(2−x−y) is an invariant probability measure for S.
Proof. It is easy to see that the projection of η to X is an invariant measure for S. Then we have
Z
(−∞,0)×(−∞,0)
1
|(x+y)−(z+w)|3dzdw+ Z
(1,∞)×(1,∞)
1
|(x+y)−(z+w)|3dzdw
= 1
(x+y)(2−x−y).
This is the assertion of this corollary.
We can compute the entropyHη(S) explicitly from Theorem 9.12 and Corollary 10.2.
Proposition 10.3.
Hη(S) = π2 8 log 2.
Proof. From Proposition 5.3 and Corollary 10.2, we complete this lemma.
From this proposition, we obtain the exponential divergence of q(k)3 ask → ∞.
Proposition 10.4.
k→∞lim 1
klogq3(k) = π2 24 log 2 for λ-a.e. (x, y).
Proof. From the Shannon-MacMillan-Breiman theorem, we have
− lim
k→∞
1
klogη(∆k) = π2
8 log 2 η-a.e.
where ∆kis defined by (εi, ni, mi) = (εi(x, y), ni(x, y), mi(x, y)) for 1≤i≤k. We take (x, y) so that h(S(x, y, z, w))· |detD(S(x, y, z, w))| ·h−1(x, y, z, w) = 1 for h(x, y, z, w) = dη/dλ holds. Then we choose a subsequence ((lk) : k ≥1) by
l1 = min{l ≥1|(εl(x, y), nl(x, y), ml(x, y))6= (±1,1,1)}
and
lk+1 = min{l > lk |(εl(x, y), nl(x, y), ml(x, y))6= (±1,1,1)
or (εlk+1, nlk+1, mlk+1) = (+1,1,1), (εlk, nlk, mlk)6= (+1,1,1), (εlk+1, nlk+1, mlk+1) = (−1,1,1), (εlk, nlk, mlk)6= (−1,1,1)}.
for k ≥1, which means that we choose all cylinders ∆l ∈R(S). Since ∆l is bounded away from (0,0) and (1,1), there exists a constant C1 >1 such that
1 C1
λ(∆lk) < η(∆lk) < C1λ(∆lk).
On the other hand, there exists a constant C2 >1 and C20 >1 such that 1
C2q3(l) < λ(∆l) < C2
q3(l) for εl = +1 1
C20(2q(l)1 +q3(l)) < λ(∆l) < C20
(2q(l)1 +q3(l)) for εl =−1
whenever ∆l ∈ R(S), see Lemma 9.6. But, if ∆l ∈ R(S), we see that 3|q(l)1 | < q3(l) for εl =−1 from the proof Lemma 9.2. Then there exists a constant C3 >1 such that
1
C3q3(l) < λ(∆l) < C3
q3(l) whenever ∆l∈R(S). Hence we obtain
k→∞lim 1
lklogq(l3k) = π2 24 log 2
for η-a.e. (x, y). It is clear thatq3(k)=q(k−1)3 if (εk(x, y), nk(x, y), mk(x, y)) = (+1,1,1) and 2q(k)1 +q3(k) = 2q(k−1)1 +q(k−1)3 if (εk(x, y), nk(x, y), mk(x, y)) = (−1,1,1). Since the indicator function of h(±1,1,1)i is obviously integrable with respect to η,
k→∞lim
lk−lk−1 lk = 0 for η-a.e. (x, y). Hence we have
l→∞lim 1
l logq3(l) = π2 24 log 2 for η-a.e. (x, y), equivalently λ-a.e.
11 Characterization of periodic points of the modified negative slope algorithm
In the previous section, we define S, the natural extension of the modified negative slope algorithm, in X = [0,1]2 × {(−∞,0)∪(1,∞)2}. In this section, we show the following theorem.
Theorem 11.1. Suppose iteration by the modified negative slope algorithmS of (x, y)∈X does not stop. Then the sequence (Sk(x, y) : k ≥ 0) is purely periodic if and only if x and y are in the same quadratic extension of Q and (x, y, x∗, y∗)∈X where x∗ denotes the algebraic conjugate of x.