Distribution of preperiodic points in one-parameter families of rational maps

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript t"> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">f_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a one-parameter family of rational maps defined over a number field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> outside of a set of natural density zero, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -rational preperiodic point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript t"> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">f_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the specialization of some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -rational preperiodic point of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -rational preperiodic points of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript t Baseline left-parenthesis z right-parenthesis equals z squared plus t"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>t</mml:mi> </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>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f_t(z) = z^2 + t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over the field of rational numbers.