Polynomial entropy and rotation sets

Abstract

<jats:p> We investigate the polynomial entropy of continuous maps on the <jats:italic>d</jats:italic> -dimensional torus <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\mathbb {T}^d$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> ( <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d \ge 1$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> ) that are homotopic to the identity. We establish that if the associated rotation set of such a map contains more than a single point, then its polynomial entropy is at least 1. As a corollary, we show that for any orientation-preserving homeomorphism <jats:italic>f</jats:italic> on the 2-dimensional torus with polynomial entropy strictly less than 1,  there exists some <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$q \in \mathbb {N}$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> such that the iterate <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f^q$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is a pseudo-rotation; that is, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$f^q$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is homotopic to the identity and its rotation set is a singleton. </jats:p>