arXiv subjects

William Gignac

Publications and source records attributed to William Gignac.

5 recordsLinked to original sources

arXiv2018

Local dynamics of non-invertible maps near normal surface singularities

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs $f\colon (X,x_0)\to (X,x_0)$, 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 $π\colon X_π\to (X,x_0)$ such that the dynamics of $f$ (or more precisely of $f^N$ for $N$ big enough) on $X_π$ 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.

math.DS↗

arXiv2014

On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold

We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.

math.DS↗

arXiv2013

Growth of attraction rates for iterates of a superattracting germ in dimension two

We study the sequence of attraction rates of iterates of a dominant superattracting holomorphic fixed point germ f:(C^2,0)->(C^2,0). By using valuative techniques similar to those developed by Favre-Jonsson, we show that this sequence eventually satisfies an integral linear recursion relation, which, up to replacing f by an iterate, can be taken to have order at most two. In addition, when the germ f is finite, we show the existence of a bimeromorphic model of (C^2,0) where f satisfies a weak local algebraic stability condition.

math.DS↗

arXiv2013

Equidistribution of preimages over nonarchimedean fields for maps of good reduction

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages of any tame valuation equidistribute to a canonical measure.

math.DS↗

arXiv2012

Measures and dynamics on Noetherian spaces

We give an explicit description of all finite Borel measures on Noetherian topological spaces X, and characterize them as objects dual to a space of functions on X. We use these results to study the asymptotic behavior of continuous dynamical systems on Noetherian spaces.

math.DS↗