The dynamical degree of a dominant rational map $f:\mathbb{P}^N\to\mathbb{P}^N$ is defined by the limit $$ \delta(f) := \lim_{n\to\infty} \deg(f^{\circ n}), $$ where $f^{\circ n}$ is the $n$th iterate of $f$. It was conjectured by Bellon and Vialet [1] that $\delta(f)$ is always an algebraic integer, and over the succeeding two decades, this was proven for many classes of maps. But Bell, Diller, and Jonsson [2] recently gave an example of a map on $\mathbb{P}^2$ whose dynamical degree is a transcendental number.

[1] Algebraic entropy, Bellon, M. P. and Viallet, C.-M., Comm. Math. Phys. 204 (1999), 425-437.

[2] A transcendental dynamical degree, Jason P. Bell, Jeffrey Diller, Mattias Jonsson, https://arxiv.org/abs/1907.00675

The dynamical degree of a dominant rational map $f:\mathbb{P}^N\to\mathbb{P}^N$ is defined by the limit $$ \delta(f) := \lim_{n\to\infty} \deg(f^{\circ n})^{1/n}, $$ where $f^{\circ n}$ is the $n$th iterate of $f$. It was conjectured by Bellon and Vialet [1] that $\delta(f)$ is always an algebraic integer, and over the succeeding two decades, this was proven for many classes of maps. But Bell, Diller, and Jonsson [2] recently gave an example of a map on $\mathbb{P}^2$ whose dynamical degree is a transcendental number.

[1] Algebraic entropy, Bellon, M. P. and Viallet, C.-M., Comm. Math. Phys. 204 (1999), 425-437.

[2] A transcendental dynamical degree, Jason P. Bell, Jeffrey Diller, Mattias Jonsson, https://arxiv.org/abs/1907.00675