The calculated Misiurewicz point is
A
= 3.67857351042832226... at
X
= 0.728155493653961819...
According to Daniel Stoffer of the Mathematics Department at
ETH-Zurich,
A satisfies the
equation
A^{3} - 2
A^{2} - 4
A - 8 = 0, and
X follows from
X = 1 - 1/
A. He also points out that the
Misiurewicz point has the following properties:
- For any A less than or
equal to the Misiurewicz point, the logistic map admits no periodic
point of odd period except for a fixed point. For any A greater than the Misiurewicz
point, the logistic map admits periodic points of odd period > 1.
For A > 3.628 but less
than or equal to the Misiurewicz point, the logistic map admits
periodic points of any even period.
- If A is less than or
equal to the Misiurewicz point, then the logistic map does not admit a
snap-back repellor. (For the notion of a snap-back repellor, see
F. Marotto, Snap-back repellors
imply chaos in R^{n}, J. Math. Anal. Appl. (1978).)
Ref: M. Misiurewicz,
Absolutely
continuous
measures
for
certain maps of
an interval, Publ. Math. I.H.E.S.
53,
17-51
(1981).