A Note on the Growth of Periodic Points for Commuting Toral Automorphisms

<table>The projection of <svg height="12.4625" id="M171" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 19.293751 12.4625" width="19.293751" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.4625)">
<text transform="matrix(1,0,0,-1,0.498,3.72)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑤</tspan>
</text>
<text transform="matrix(1,0,0,-1,10.188,0.594)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝜃</tspan>
</text>
</g>
</svg> onto one of the vectors <svg height="12.31" id="M172" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 15.265 12.31" width="15.265" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.31)">
<text transform="matrix(1,0,0,-1,0.498,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,6.825,0.498)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
</g>
</svg>, <svg height="12.31" id="M173" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 15.265 12.31" width="15.265" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.31)">
<text transform="matrix(1,0,0,-1,0.498,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,6.825,0.498)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</svg>, <svg height="12.46375" id="M174" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 15.265 12.46375" width="15.265" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.46375)">
<text transform="matrix(1,0,0,-1,0.498,3.746)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
</text>
<text transform="matrix(1,0,0,-1,6.825,0.621)">
<tspan style="font-size: 8.75px; " x="0" y="0">3</tspan>
</text>
</g>
</svg> must have a strictly positive component. </table>

International Scholarly Research Notices

fig3

Figure 3

Figure 3: A Note on the Growth of Periodic Points for Commuting Toral Automorphisms 