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Some interesting birational morphisms of smooth affine quadric $3$-folds

Cinzia Bisi, Jonathan D. Hauenstein, Tuyen Trung Truong

TL;DR

The paper analyzes a family of birational maps on smooth affine quadric 3-folds derived from a polynomial map on C^4 that preserves the fibres of phi = x_1 x_4 − x_2 x_3, focusing on the restricted dynamics on Z_c and its projective closure X_c. It demonstrates, through a blend of rigorous theory and Bertini-based numerical experiments, that the first dynamical degree matches the ambient growth zeta_1 ≈ 1.8393 while the second dynamical degree lambda_2(hat f_c) equals the largest root zeta_2 ≈ 2.1108 of a cubic 2t^3 − 3(t^2 − 1) − 4t and is not an algebraic integer, suggesting a potential counterexample to conjectures about equidistribution and algebraicity of dynamical degrees. The work also provides extensive experimental data on fixed points and degree sequences, proposes a roadmap with concrete questions to bound lambda_2 from above and below, and discusses the implications for the existence and growth of isolated periodic points, highlighting the need for new techniques beyond current methods. Overall, the paper contributes a concrete, computationally informed case that challenges established beliefs about dynamical degrees and periodic point growth in higher-dimensional birational dynamics.

Abstract

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are cohomologically hyperbolic, b) their second dynamical degree is an algebraic number but not an algebraic integer, and c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on $\mathbb{C}^4$ preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

Some interesting birational morphisms of smooth affine quadric $3$-folds

TL;DR

The paper analyzes a family of birational maps on smooth affine quadric 3-folds derived from a polynomial map on C^4 that preserves the fibres of phi = x_1 x_4 − x_2 x_3, focusing on the restricted dynamics on Z_c and its projective closure X_c. It demonstrates, through a blend of rigorous theory and Bertini-based numerical experiments, that the first dynamical degree matches the ambient growth zeta_1 ≈ 1.8393 while the second dynamical degree lambda_2(hat f_c) equals the largest root zeta_2 ≈ 2.1108 of a cubic 2t^3 − 3(t^2 − 1) − 4t and is not an algebraic integer, suggesting a potential counterexample to conjectures about equidistribution and algebraicity of dynamical degrees. The work also provides extensive experimental data on fixed points and degree sequences, proposes a roadmap with concrete questions to bound lambda_2 from above and below, and discusses the implications for the existence and growth of isolated periodic points, highlighting the need for new techniques beyond current methods. Overall, the paper contributes a concrete, computationally informed case that challenges established beliefs about dynamical degrees and periodic point growth in higher-dimensional birational dynamics.

Abstract

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form constant, which seems to have some (among many others) interesting/unexpected characters: a) they are cohomologically hyperbolic, b) their second dynamical degree is an algebraic number but not an algebraic integer, and c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.
Paper Structure (13 sections, 8 theorems, 53 equations)

This paper contains 13 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Method
  3. Results
  4. Theoretical results
  5. Experimental results
  6. Further analysis $\&$ A road map towards an affirmative answer to Conjecture \ref{['Conjecture3']}
  7. An approach towards establishing the upper bound $\lambda _2(\widehat{f}_c)\leq \zeta _2$
  8. Two approaches towards establishing the lower bound $\lambda _2(\widehat{f}_c)\geq \zeta _2$
  9. An approach to part 3 of Conjecture \ref{['Conjecture3']}
  10. A speculation on the exponential growth of the isolated periodic points
  11. Discussion
  12. Conclusions
  13. Acknowledgements and Declarations

Key Result

Proposition 2.1

Let $X\subset \mathbb{P}^{2m}$ be a smooth quadric. Let $g:X\dashrightarrow X$ be a dominant rational map. Then the Lefschetz numbers $L(g^n)$ (i.e. the intersection number between the graph of $g^n$ and the diagonal of $X$) are all positive, and satisfy log concavity, i.e. for all $n,n'\geq 0$. In particular, the sequence $\{L(g^n)^{1/n}\}_{n=1,2,\ldots}$ is decreasing. Similarly, for every $0\l

Theorems & Definitions (23)

  • Conjecture 1.1: Folklore Conjecture
  • Conjecture 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 13 more