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This Mandelbrot set is constructed from two conjugate critical points, and if we let z1 and z2 be the images of these by (1 - z^2)/(z - z^2cos(z)), we have that if c is real and numerical large and belonging to the Mandelbrot set, then z1 + c and z2 + c belong to different Fatou domains for (1 - z^2)/(z - z^2cos(z)) + c, therefore 1/cos(z1 + c) + c and 1/cos(z2 + c) + c belong to different Fatou domains, and if we add a multiple of 2π to c (so that the result still is numerically large), then these two numbers will belong to different Fatou domains. Therefore we can say: it is possible that the Mandelbrot set extends towards infinity in the horizontal direction. It seems really to be the case, as we can see if we draw the Mandelbrot set and zoom out.
The two following pictures show sections of the Julia set for c = -1. The point z = -1 is a fixed point for this iteration, and as the derivative of the function in this point is numerically larger than 1, z = -1 is repelling and belongs therefore to the Julia set. If x is real and numerically very large, the iteration of x is near 1/cos(x) - 1. Therefore, if x* is a real point of the Julia set such that x* - (-1) > 1, we can find a real point of large numerical value that iterates into this point of the Julia set, and this indicates that the Julia set extends towards infinity in the horizontal direction. And as cos(z) grows onentially in the vertical direction, the iteration of a point z of large numerical y-value is very near -1, therefore we can find points of arbitrary large y-values that iterate into -1, so that the Julia set extends towards infinity in the vertical direction. |
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