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Resurgence of receeding limbs

Carsten Lunde Petersen
Torsdag, 28 april, 2011, at 16:15, in Aud. D3 (1531-215)

The famous Mandelbrot set $M$ is the set of complex parameters $c$ for which the quadratic polynomial $z^2+c$ has a connected Julia set. These polynomials are dynamically characterized (up to Moebius-conjugacy) as those quadratic rational maps which have a super attracting (i.e. critical) fixed point (the point infinity). When we deform the quadratic polynomials transversely to the complex line of polynomials within the space of quadratic rational maps, then the fixed point at infinity and the critical point at infinity separate to a simple critical point and a nearby attracting fixed point of multiplier initially close to zero. When we vary this multiplier over $D = \{ |z|<1 \}$, this will define a holomorphic motion of $M$. The natural boundary for this motion is the unit circle $S$.

This raises questions like: Are there any limits of the motion as the parameter gets near $S$ say radially. If/when such limits exists, do they admit tangible descriptions? This is in particular interesting for roots of unity. This is where "rescalings make limbs resurge".

Kontaktperson: Bent Ørsted