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Journal of Modern Dynamics (JMD)
 

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

Pages: 541 - 561, Volume 10, 2016      doi:10.3934/jmd.2016.10.541

 
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Hieu Trung Do - Department of Mathematics, Oregon State University, Corvallis, OR 97331, United States (email)
Thomas A. Schmidt - Department of Mathematics, Oregon State University, Corvallis, OR 97331, United States (email)

Abstract: We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give a new infinite family of such maps in genus 3.

Keywords:  Pseudo-Anosov, Sah-Arnoux-Fathi invariant, Pisot units, bi-Perron units, translation surfaces.
Mathematics Subject Classification:  Primary: 37E30, 57M50; Secondary: 11R06.

Received: March 2016;      Revised: September 2016;      Available Online: November 2016.

 References