Misiurewicz point

The Misiurewicz point (also Misiurewicz-Thurston point) is named after the Polish mathematician Misiurewicz . Such a point is calculated in order to demonstrate the similarity of a connected Julia set to the boundary of the Mandelbrot set for the same Misiurewicz point in a graphical representation. In a publication on the similarity of the Mandelbrot set and Julia set, Tan Lei showed that the representation of the Mandelbrot set at a Misiurewicz point, apart from a magnification factor and a rotation, is a deformed image of the Julia set on the same Misiurewicz -Point is.

Furthermore, Misiurewicz points are used for the graphical representation of the self-similarity of the Mandelbrot set, Multibrot set and for fractals .

definition

The following definition of the Misiurewicz point can be found in the literature:

The parameter value is just then a Misiurewicz-point when the präperiodische Orbit opens into a periodic orbit. ${\displaystyle c_{0}}$

This definition is based on properties of a recursive sequence, which are explained below.

For a complex quadratic polynomial a recursion is given in the representation . The start value is a fixed initial value and the complex parameter is a freely selectable variable. With these specifications, the recursive sequence has the following form: ${\displaystyle f_{c}\colon {\overline {\mathbb {C} }}\mapsto {\overline {\mathbb {C} }},\,z\mapsto z^{2}+c}$ ${\displaystyle z_{0}=0}$ ${\displaystyle c}$

${\displaystyle f_{c}^{0}(0)\mapsto f_{c}^{1}(0)\mapsto f_{c}^{2}(0)\mapsto f_{c}^{3}(0)\mapsto \cdots \mapsto f_{c}^{n}(0)}$.

Here and means the n-times consecutive execution of and must not be understood as the n-th power. ${\displaystyle f_{c}^{0}=z_{0}}$ ${\displaystyle f_{c}^{n}}$ ${\displaystyle f_{c}}$

Now let the complex parameters for further calculation to the value set and the abbreviation, where it makes sense . Then the recursive sequence for the -th and -th consecutive execution, under the condition that a Misiurewicz point is present, has the representation: ${\displaystyle c}$ ${\displaystyle c_{0}}$ ${\displaystyle f_{c}^{l}(c_{0})=\alpha }$ ${\displaystyle l}$ ${\displaystyle p}$

${\displaystyle 0\mapsto c_{0}\mapsto f_{c}(c_{0})\mapsto \cdots \mapsto f_{c}^{l}(c_{0})=\alpha \mapsto f_{c}(f_{c}^{l}(c_{0}))=f_{c}(\alpha )\mapsto \cdots \mapsto f_{c}^{p}(\alpha )=f_{c}(\alpha )\mapsto \cdots }$.

The properties of this sequence can be summarized as follows:

• A pre-periodic orbit is generated up to the th sequential element. The pre-periodic orbit has the representation and it applies because it must be part of the pre-periodic orbit. ${\displaystyle l}$ ${\displaystyle \{0,c_{0},f_{c}(c_{0}),\cdots ,f_{c}^{l}(c_{0})\}}$ ${\displaystyle l\geq 1}$ ${\displaystyle c_{0}}$
• A cyclic orbit arises from the -th successor element and therefore must be. ${\displaystyle p}$ ${\displaystyle \gamma =\{f_{c}^{p}(\alpha ),\cdots ,f_{c}^{2p-1}(\alpha )\}}$ ${\displaystyle p\geq 1}$
• By means of induction it can be shown that holds for anything . ${\displaystyle f_{c}^{k\cdot p}(\alpha )=f_{c}^{p}(\alpha )}$ ${\displaystyle k\in \mathbb {N} }$

Examples

• For the starting value is obtained for with the recursive sequence: ${\displaystyle c=i}$ ${\displaystyle f_{i}\colon z\mapsto z^{2}+i}$ ${\displaystyle z_{0}=0}$

${\displaystyle 0\mapsto i\mapsto i-1\mapsto -i\mapsto i-1\cdots \mapsto i-1\mapsto -i\mapsto i-1\cdots }$.

The pre-periodic orbit is and ends in a periodic orbit . Hence there is a Misiurewicz point. ${\displaystyle \{0,i\}}$ ${\displaystyle \gamma =\{i-1,-i\}}$ ${\displaystyle i}$

• With a start value , the recursive sequence for with is : ${\displaystyle c=-{\tfrac {1}{2}}}$ ${\displaystyle f_{-1/2}\colon z\mapsto z^{2}-{\tfrac {1}{2}}}$ ${\displaystyle z_{0}=0}$

${\displaystyle 0\mapsto -{\tfrac {1}{2}}\mapsto -{\tfrac {1}{4}}\mapsto -{\tfrac {7}{16}}\mapsto -{\tfrac {79}{256}}\mapsto \cdots }$.

${\displaystyle c=-{\tfrac {1}{2}}}$ is not a Misiurewicz point, because there is no pre-periodic orbit and no periodic orbit.

literature

• Dierk Schleicher: On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets, in: M. Lapidus, M. van Frankenhuysen (eds): Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics 72, American Mathematical Society (2004), 477-507, 1999, pdf

Individual evidence

1. ^ Tan Lei: Similarity between the Mandelbrot set and the Julia sets, Communications in Mathematical Physics, Vol 134 Number 3, pp. 587-617, 1990, pdf
2. ↑ Dierk Schleicher: rational Parameter Rays of the Multibrot Sets, 2015, pdf
3. ↑ Dierk Schleicher: rational Parameter Rays of the Multibrot Sets, page 30, 2015, pdf