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A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\epsilon' \ge \epsilon_x$ and a similarity $s$ of $E$ such that $$s(F \cap B(y,\epsilon')) = F \cap B(x,\epsilon) $$ with $B(z,r)$ the ball of center $z$ and radius $r$.

Question: Is the Mandelbrot set weakly self-similar?

If so: Is there $\alpha>0$ such that for all $x \in F$, we can take $\epsilon_x>\alpha ?$

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  • $\begingroup$ @ClaudeChaunier: in the definition written, weakly self-similar does not main self-similar with a weak similarity (sorry for the confusion). If $\epsilon < \epsilon_x$ then $\epsilon < \epsilon'$, and so $s$ is not an isometry. $\endgroup$ – Sebastien Palcoux Nov 11 '18 at 19:16
  • $\begingroup$ Sorry, there is no confusion in your definition, I should have seen that the part around $y$ is required to be bigger. Thank you to have pointed to it. $\endgroup$ – Claude Chaunier Nov 12 '18 at 6:16

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