Parabolic tools

Abstract

In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in super-attracting basins. It is used to construct parabolic puzzles similar to Yoccoz puzzles, but modified near the parabolic fixed point. As an application, we show that any non-renormalizable quadratic rational map in the connectedness locus M ₁ of Per₁(1) has locally connected Julia set.

Keywords:

• parabolic rays
• parabolic Yoccoz puzzle
• parabolic Mandelbrot set M ₁
• Yoccoz' combinatorial analytic invariant

AMS Subject Classification::

• Primary: 37F10
• 37F20
• 37F45. Secondary: 30D05
• 30D40
• 32H50

Acknowledgements

The first author would like to thank Institut de Mathématiques de Toulouse, Université Paul Sabatier, the Research Council for Nature and Universe, Denmark through the grant 272-07-0321 and the EU through the Marie Curie RTN network CODY, grant nr 35651-2 for support. The second author was partially supported by the grant ABC of the Agence Nationale de la Recherche, France.

Notes

1. [email protected]

Additional information

Notes on contributors

Pascale Roesch

1