2025/08/20 by Kim, Kyounghee, Park, Insung
#14E07 #20F28 #20F34 #37E30 #57S05 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2508.14728
We study the real dynamics of a family of rational surface automorphisms obtained from quadratic birational maps of \pcc that preserve a cuspidal cubic and whose critical orbits have lengths (1,m,n) with 1+m+n≥ 10. Passing to the real locus and cutting along the invariant cubic, we obtain a diffeomorphism of an orientable surface whose fundamental group is free. Our key device is a finitely generated invariant, positive semigroup Sm,n in the fundamental group on which an iterate of induced action acts by concatenation without cancellation. This positivity yields a nonnegative primitive transition matrix, so Perron-Frobenius theory supplies an explicit exponential growth rate λ>1 for the induced action on the fundamental group. Consequently, the real map has positive topological entropy. We package the combinatorics of the generators in a ``Core-Tail Induction Principle," which allows us to treat simultaneously seven orbit-data families with only finite base checks. Finally, using Bestvina-Handel and the Dehn-Nielsen-Baer correspondence, we show that the induced outer automorphism with m+n odd is realized by a pseudo-Anosov homeomorphism of the cut surface.