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Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not)

Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family. I wondered is it known that the MLC is not enough (or ...
yaoxiao's user avatar
  • 1,706
3 votes
0 answers
80 views

Confusion on the assumption when discussing the kneading invariants for unimodal maps

A unimodal map is a continuous map $f:[0,1]\longrightarrow [0,1]$ such that there is only one turning point (critical point), denoted by $c$, and $f(0)=f(1)=0$. Unimodal map is related to kneading ...
JacobsonRadical's user avatar
2 votes
0 answers
89 views

Conjugacy between piecewise linear circle maps

Let $\mathcal{M}$ the Mandelbrot set, $\mathcal{M}=\{c \in \mathbb{C}: \{Q_c^n(0) \}_{n \in \mathbb{N}} \text{ is bounded, where } Q_c(z)=z^2+c \}$ And let the hyperbolic or stable component, $H_n=\{ ...
confused's user avatar
  • 271
2 votes
0 answers
79 views

When is replacing the prefix of an angled internal address a valid operation?

While working on an artwork exploring patterns in the Mandelbrot set fractal, I constructed an angled internal address by: $$ 1 \overset{1/2}\longrightarrow 2 \overset{1/2}\longrightarrow 3 \overset{1/...
Claude's user avatar
  • 111