From hyperbolic to parabolic parameters along internal rays

<p>For the quadratic family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript c Baseline left-parenthesis z right-parenthesis equals z squared plus c"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f_{c}(z) = z^2+c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to a parabolic parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove c With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>c</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\hat {c}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove c With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>c</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\hat {c}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove c With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>c</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\hat {c}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with exponent depending only on the petal number.</p> <p>This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].</p>