A birational map of a projective space whose intermediate dynamical degrees are all transcendental

TL;DR

The paper answers whether there exists a birational map on projective space with all intermediate dynamical degrees transcendental for $d\ge6$ by constructing a base example on $\mathbb{P}^3$ with a transcendental $\lambda_1$ and an algebraic $\lambda_2$, and then combining products, birational conjugacies, and a dimension-raising embedding to produce a map on $\mathbb{P}^6$ with $\lambda_1,\dots,\lambda_5$ all transcendental. It then uses an induction step to lift the construction to all $d\ge6$, preserving the transcendence of each intermediate degree. This provides a positive answer to a question of BDJK24 about simultaneous transcendence of all intermediate degrees and demonstrates a systematic method to control the entire spectrum of dynamical degrees through birational and product operations. The results highlight intricate interactions between algebraic and transcendental dynamical invariants in higher-dimensional birational dynamics with potential implications for the study of dynamical systems on projective varieties.

Abstract

We construct a birational map of $\mathbb{P}^d$ ($d\geq6$) whose intermediate dynamical degrees are all trancendental.

Figures (2)

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Theorems & Definitions (14)

• Theorem 1.1: DF01
• Theorem 1.2: cf. BDJK24
• Remark 1.3
• Theorem 1.4: cf. Sug24
• Theorem A
• Proposition 2.1
• proof : Proof of Proposition \ref{['inverse map']}
• Proposition 2.2: cf. DS05, DN11, Tru15
• Proposition 2.3: cf. Tru15, DN11
• Proposition 3.1