Capacities and their applications to complex dynamical systems

The linearization problem is a fundamental question in local complex dynamics. Let $f$ be a holomorphic germ at the origin in $\mathbb C^n$ with $Df(0)=\operatorname{diag}(\lambda_1,\ldots,\lambda_n).$ We say that $f$ is linearizable if it is holomorphically conjugate to its linear part in a neighbourhood of the origin. A remarkable feature of this problem is that linearizability is governed largely by the arithmetic properties of the multiplier $\lambda=(\lambda_1,\ldots,\lambda_n).$
Let $\mathcal B_n$ denote the set of multipliers for which the corresponding germ is always linearizable. In the one-dimensional case, Sadullaev and Rakhimov proved that the exceptional set $\mathbb C\setminus \mathcal B_1$ is small in the sense of capacity. In this Phd thesis, we refine their result in dimension $n=1$ and establish a higher-dimensional analogue. We also develop related capacity estimates for other exceptional sets arising in complex dynamical systems.