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We construct a $1$-cohomologically hyperbolic birational map of$\mathbb{P}^3$, with transcendental first dynamical degree. The arithmeticdegree of this map at a $\overline{\mathbb{Q}}$-point is transcendental.

A $1$-cohomologically hyperbolic birational map of $\mathbb{P}^3$, with a transcendental arithmetic degree