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Ergodic Theory. Homework 4.
(due ∼ 02.25.2015)
1. Prove that the product of irrational rotations Rα1 , . . . Rαn is ergodic if
and only if α1 , . . . , αn are indeendent over rational numbers.
2. Let ∆1 , . . . , ∆n , . . . are the intervals constructed by the induced transformations from the rigid rotation by angle α having continued fraction
|∆n |
= an .
[0; a1 , a2 , a3 , . . . ]. Prove that
|∆n+1 |
3. Let α = [a0 ; a1 , a2 , . . . ]. Denote
pn
= [a0 ; a1 , . . . an ].
qn
• Prove that
pn pn−1
a0 1
a1 1
an 1
=
·
···
qn qn−1
1 0
1 0
1 0
• Prove
that qn and
n satisfy the following properties: 1) qn >
√ pn−2
√ n−2
( 2) , pn > ( 2) , 2) pn qn−1 − pn−1 qn = (−1)n+1 , 3)
|α − pqnn | < qn q1n+1 .
Some experiments.
4.
• Construct
a continued fraction for the number
√
1 + ( 2 − 1) and (a − b)(a + b) = a2 − b2 ).
√
√
2. (Hint 2 =
√
5−1
.
2
• Prove that if for some integre numbers p and q one gets α2 + pα +
q = 0 then continued fraction for α is periodic.
• Construct a continued fraction for the number
• Prove that if continued fraction for α is periodic then α is a
quadratic irrational (i.e is a solution of the equation aα2 + bα + c = 0
for some integer a, b and c)
5. First terms in the continueds fraction for π is [3; 7] which provides
22
355
π '
. Next term is 15 which gives approximant
. Find two
7
113
next terms. How does the approximation error behave? What are the
continued fraction for e?
1