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Local Dynamics of Non-Invertible Maps Near Normal Surface Singularities

William Gignac
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      We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X, x0) --> (X, x0), where X is a complex surface having x0 as a normal singularity. We prove that as long as x0 is not a cusp singularity of X, then it is possible to find arbitrarily high modifications ?: X? --> (X, x0) such that the dynamics of f (or more precisely of fN for N big enough) on X? is algebraically stable. This result is proved by understanding the dynamics induced by f on a space of valuations associated to X; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.
      Format: Paperback / softback CONTRIBUTORS: William Gignac EAN: 9781470449582 COUNTRY: United States PAGES: WEIGHT: 220 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-05-01 CITY: GENRE: MATHEMATICS / Mathematical Analysis WIDTH: 178 cm SPINE:

      Book Themes:

      Calculus and mathematical analysis

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      William Gignac, University of Michigan, Ann Arbor, MI.Matteo Ruggiero, University of Torino, Italy.
      We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs f : (X, x0) --> (X, x0), where X is a complex surface having x0 as a normal singularity. We prove that as long as x0 is not a cusp singularity of X, then it is possible to find arbitrarily high modifications ?: X? --> (X, x0) such that the dynamics of f (or more precisely of fN for N big enough) on X? is algebraically stable. This result is proved by understanding the dynamics induced by f on a space of valuations associated to X; in fact, we are able to give a strong classification of all the possible dynamical behaviors of f on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of f . Finally, we prove that in this setting the first dynamical degree is always a quadratic integer.
      Format: Paperback / softback CONTRIBUTORS: William Gignac EAN: 9781470449582 COUNTRY: United States PAGES: WEIGHT: 220 g HEIGHT: 254 cm
      PUBLISHED BY: American Mathematical Society DATE PUBLISHED: 2022-05-01 CITY: GENRE: MATHEMATICS / Mathematical Analysis WIDTH: 178 cm SPINE:

      Book Themes:

      Calculus and mathematical analysis

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      William Gignac, University of Michigan, Ann Arbor, MI.Matteo Ruggiero, University of Torino, Italy.

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