Hausdorff dimension and conformal measures of Feigenbaum Julia sets
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Bibliographic record
Abstract
We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the “hairiness phenomenon”, there exist many Feigenbaum Julia sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript normal c normal r"> <mml:semantics> <mml:msub> <mml:mi> δ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">c</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\delta _{\mathrm {cr}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equal to the hyperbolic dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper H normal upper D Subscript normal h normal y normal p Baseline left-parenthesis upper J left-parenthesis f right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> <mml:mi mathvariant="normal">y</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {HD}_{\mathrm {hyp}}(J(f))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a r e a upper J left-parenthesis f right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>area</mml:mi> <mml:mo> </mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {area} J(f)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper D Subscript normal h normal y normal p Baseline left-parenthesis upper J left-parenthesis f right-parenthesis right-parenthesis equals upper H upper D left-parenthesis upper J left-parenthesis f right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>HD</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> <mml:mi mathvariant="normal">y</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>HD</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {HD}_{\mathrm {hyp}} (J(f))=\operatorname {HD}(J(f))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the stationary case, the last statement can be reversed: if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a r e a upper J left-parenthesis f right-parenthesis greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>area</mml:mi> <mml:mo> </mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {area} J(f)> 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H upper D Subscript normal h normal y normal p Baseline left-parenthesis upper J left-parenthesis f right-parenthesis right-parenthesis greater-than 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>HD</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">h</mml:mi> <mml:mi mathvariant="normal">y</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it