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A note on pseudo-Anosov maps with small growth rate. (English) Zbl 1049.37029
From the introduction: We present an explicit sequence \(\varphi_k\) of pseudo-Anosov maps of surfaces of genus \(2k\) whose growth rates converge to one. This answers a question of Joan Birman, who had previously asked whether such growth rates are bounded away from one. Norbert A’Campo, Mladen Bestvina, and Klaus Johannson independently communicated this question to me. C. T. McMullen [Ann. Sci. Éc. Supér., IV. Sér. 33, 519–560 (2000; Zbl 1013.57010)] previously obtained a similar result using quite different techniques. The results of this paper grew out of massive computer experiments with my software package XTrain in the context of the REU program at the University of Illinois at Urbana-Champaign.

MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
57M60 Group actions on manifolds and cell complexes in low dimensions
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
Software:
XTrain
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References:
[1] DOI: 10.1007/BF02701770 · Zbl 0960.57008
[2] DOI: 10.2307/2946562 · Zbl 0757.57004
[3] DOI: 10.1016/0040-9383(94)E0009-9 · Zbl 0837.57010
[4] Brinkmann Peter, Experiment. Math. 9 (2) pp 235– (2000)
[5] Brinkmann Peter, Experiment. Math. 10 (4) pp 571– (2001)
[6] McMullen Curtis T., Ann. Sci. École Norm. Sup. 33 (4) pp 519– (2000)
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