A Misiurewicz point
is a point
in the parameter space
of a family of discrete dynamical systems
(indexed by a real or complex parameter) such that a critical point
is pre-periodic i.e. there are positive integers k
where is a critical point of .
Misiurewicz points are named after mathematician Michał Misiurewicz
A complex quadratic polynomial
has only one critical point. By a suitable conjugation
any quadratic polynomial can be transformed into a map of the form
which has a single critical point at
. The Misiurewicz points of this family of maps are the roots
of the equations
- k is the pre-period
- n is the period
- denotes the n-fold composition of with itself i.e. the nth iteration of .
For example, the Misiurewicz points with k=2 and n=1, denoted by M2,1, are roots of
The root c=0 is not a Misiurewicz point because the critical point is a fixed point when c=0, and so is periodic rather than pre-periodic. This leaves a single Misiurewicz point M2,1 at c = −2.
Misiurewicz points are boundary points = All Misiurewicz points belong to the boundary
of the Mandelbrot set
. Misiurewicz points are dense
at the boundary
of Mandelbrot set
If is a Misiurewicz point, then the associated filled Julia set is equal to Julia set, it means filled Julia set has no interior.
If is a Misiurewicz point, then the corresponding periodic cycle is repelling.
Mandelbrot set and Julia set are locally similar around Misiurewicz points. Mandelbrot set is self-similar around Misiurewicz points
Misiurewicz points can be :
- branch points (= points where filaments meet.) with 3 or more external arguments (angles )
- end points of :
with 1 external argument
- centers of spirals
of Misiurewicz points are rational numbers
with even denominator
, which is the end-point of main antenna of Mandelbrot set
This diagram shows iteration of critical point
of complex quadratic polynomial
Notice that it is z-plane ( dynamical plane
) not c-plane ( parameter plane
) and point
is not the same point as
One can see that critical point
is preperiodic with preperiod 2 and period 1.
of critical point
= C L R R R ...
is landing point of only one external ray
(parameter ray) of angle 1/2 .
Point is a principial Misiurewicz point of the 1/3 limb. It has 3 external rays 9/56, 11/56 and 15/56.
Point is a Misiurewicz point which is a center of a two-arms spiral.
P(n):=if n=0 then 0 else P(n-1)^2+c;
Define a function whose roots are Misiurewicz points, and find them.
Examples of use :
- Michał Misiurewicz (1981), "Absolutely continuous measures for certain maps of an interval" Publications Mathématiques de l'IHÉS, 53 (1981), p. 17-51
- G. Pastor, M. Romera and F. Montoya (1996). "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps" Physica A, 232 (1996), 536-553
- G. Pastor, M. Romera, G. Álvarez and F. Montoya (2001). "Misiurewicz point patterns generation in one-dimensional quadratic maps", Physica A, 292 (2001), 207-230
- G. Pastor, M. Romera, G. Álvarez and F. Montoya (2003). "How to work with one-dimensional quadratic maps", Chaos, Solitons and Fractals, 18 (2003), 899-915
- M. Romera, G. Pastor and F. Montoya (1996), Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535.