FINITENESS AND PERIODICITY OF BETA EXPANSIONS -
NUMBER THEORETICAL AND DYNAMICAL OPEN
SHIGEKI AKIYAMA (NIIGATA UNIVERSITY, JAPAN)
Let y = f(x) be a positive real function. Consider a digital expansion of a real
number x in a form:
x = 0 + f(1 + f(2 + f(3 + . . . .
given by an algorithm with 0 = x , r0 = x - x and
n+1 = f-1
(rn) , rn+1 = f-1
(rn) - f-1
This is R´enyi's f-expansion (). It is the usual b-adic expansion when f(x) =
x/b and the regular continued fraction when f(x) = 1/x. For f(x) = x/ with a
non-integer > 1, it is called the -expansion.
0.2 0.4 0.6 0.8 1