A new construction of the unstable manifold for the measure-preserving H{é}non map
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the measure-preserving Hénon map with the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a hyperbolic fixed point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic p"> <mml:semantics> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main result of this paper is that the unstable mainfold of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic p"> <mml:semantics> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the iterated limit of a very simple set. Informally, <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript u Baseline left-parenthesis bold-italic p right-parenthesis equals limit Underscript n right-arrow normal infinity Endscripts upper H Superscript n Baseline left-parenthesis script upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mi>u</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:munder> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^u(\boldsymbol {p}) = \lim _{n\to \infty } H^n(\mathcal L)</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the line <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y equals negative x"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">y=-x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript u Baseline left-parenthesis bold-italic p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>W</mml:mi> <mml:mi>u</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^u(\boldsymbol {p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the unstable manifold of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic p"> <mml:semantics> <mml:mi mathvariant="bold-italic">p</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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How this classification was reachedexpand
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 | 0.004 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".