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Register a new userLocal dynamics of non-invertible maps near normal surface singularities
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We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs $fcolon (X,x_0)to (X,x_0)$, where $X$ is a complex surface having $x_0$ as a normal singularity. We prove that as long as $x_0$ is not a cusp singularity of $X$, then it is possible to find arbitrarily high modifications $picolon X_pito (X,x_0)$ such that the dynamics of $f$ (or more precisely of $f^N$ for $N$ big enough) on $X_pi$ 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.
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In this article, we show that R.H. Bings pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Beguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.
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