Dynamical degrees of affine-triangular automorphisms of affine spaces

<jats:title>Abstract</jats:title><jats:p>We study the possible dynamical degrees of automorphisms of the affine space<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385721000900_inline1.png" /><jats:tex-math>$\mathbb {A}^n$</jats:tex-math></jats:alternatives></jats:inline-formula>. In dimension<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385721000900_inline2.png" /><jats:tex-math>$n=3$</jats:tex-math></jats:alternatives></jats:inline-formula>, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385721000900_inline3.png" /><jats:tex-math>$\mathbb {A}^n$</jats:tex-math></jats:alternatives></jats:inline-formula>for some<jats:italic>n</jats:italic>, and we give the best possible<jats:italic>n</jats:italic>for quadratic integers, which is either<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385721000900_inline4.png" /><jats:tex-math>$3$</jats:tex-math></jats:alternatives></jats:inline-formula>or<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385721000900_inline5.png" /><jats:tex-math>$4$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>