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1.3: The Secondary Bulbs Symmetry

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  • Page ID
    101317

    • Robert L. Devaney
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    1.3.1.jpg
    Figure \(\PageIndex{1}\): -0.1230  0.8000  0.4000   2152ms

    Every ( m1 / n1 ) primary bulb has decoration very similar to the main bulb one (in "square" parametrisation). So for the secondary bulbs we have simply to add one more m2 / n2 rotation number ( m1 / n1 m2 / n2 ). Note that period of a secondary bulb is n1 × n2 product. In this way we can map all the M-set decorations.

    1.3.2.png
    Figure \(\PageIndex{2}\): -0.0052  0.7500  0.0500   2394ms

    You can determine n2 by the number of spokes in the largest antenna attached to a secondary decoration and the shortest spoke is located m2 = 1 revolutions in the counter-clockwise direction from the main spoke in the square limited region! But you see three spokes branching point nearby.

    For ( 1/3 1/4 ) secondary bulb we have period 3 × 4 = 12 orbit (the left image). But after factorisation (if we connect only 1, 4, 8 and 12 points) we get f o3: z1 → z4 → z8 → z12 → z1 second order star (small 4-polygon on the right image).

    1.3.3.png
    Figure \(\PageIndex{3}\): 0.0200  0.0000  2.5500   347ms
    1.3.4.png
    Figure \(\PageIndex{4}\): 0.0200  0.0000  2.5500   918ms

    At last, the Jc-set contains infinitely many "junction points" at which 4 distinct black regions in J are attached. And the smallest black region is located m1 = 1 revolutions in the counter-clockwise direction from the largest central region. Again in the square limited region.

    This page titled 1.3: The Secondary Bulbs Symmetry is shared under a Public Domain license and was authored, remixed, and/or curated by Robert L. Devaney via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.