Beschreibung:
<p>The dynamical degree <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda left-parenthesis f right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>λ<!-- λ --></mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\lambda (f)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of a birational transformation <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f">
<mml:semantics>
<mml:mi>f</mml:mi>
<mml:annotation encoding="application/x-tex">f</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> measures the exponential growth rate of the degree of the formulas that define the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>th iterate of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f">
<mml:semantics>
<mml:mi>f</mml:mi>
<mml:annotation encoding="application/x-tex">f</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda left-parenthesis f right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>λ<!-- λ --></mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\lambda (f)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and the structure of the conjugacy class of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f">
<mml:semantics>
<mml:mi>f</mml:mi>
<mml:annotation encoding="application/x-tex">f</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.</p>