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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 enough) to prove the density of hyperbolic conjecture for the family of unicritical polynomial family with degree d ($d\geq 2$), or the more general family of rational maps with degree d.

Any comments and references will be appreciated.

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I guess "MLC implies HD" holds for any unicritical polynomial family (probably the same proof applies but I haven't checked). On the other hand, Lavaurs proved in his thesis that the connectedness locus of cubic polynomial family is not locally connected, so your question does make sense only for (essentially) unicritical families.

For example, Roesch proved that the connectedness locus of some one-parameter family is locally connected outside baby Mandelbrot sets. Probably it is natural to expect that for such a family of essentially unicritical polynomials, both the local connectivity of the connectedness locus and the density of hyperbolicity are equivalent to those of the Mandelbrot set (or the Multibrot set of the corresponding degree).

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