Anosov Vector Fields and Fried Sections

Abstract The purpose of this paper is to prove that if Y is a compact manifold, if Z is an Anosov vector field on Y , and if F is a flat vector bundle, there is a corresponding canonical nonzero section $$\tau _{\nu }\left( i_{Z}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>τ</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> of the determinant line $$\nu =\det H\left( Y,F\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mo>det</mml:mo> <mml:mi>H</mml:mi> <mml:mfenced> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . In families, this section is $$C^{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> with respect to the canonical smooth structure on $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> . When F is flat on the total space of the corresponding fibration, our section is flat with respect to the Gauss-Manin connection on $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> .