Global smooth rigidity for toral automorphisms

Abstract We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> and a $C^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> diffeomorphism $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> of the torus. For a very weakly irreducible hyperbolic automorphism $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> we show that any $C^{1}$ <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> conjugacy is $C^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> . For a very weakly irreducible ergodic partially hyperbolic automorphism $L$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> we show that any $C^{1+\text{H"{o}lder}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mtext>Hölder</mml:mtext> </mml:mrow> </mml:msup> </mml:math> conjugacy is $C^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> . As a corollary, we improve regularity of the conjugacy to $C^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> in prior local and global rigidity results.