arXiv subjects
Publications and source records attributed to Michel Zinsmeister.
We show that the supremum for $c$ real of the Hausdorff dimension of the Julia set of the polynomial $z\mapsto z^d+c$ ($d$ is an even natural number) is greater than $2d/(d+1)$.
We are interested on the statistics of the duration of Brownian diffusions started at distance εfrom a given boundary and stopped when they hit back the interface.
In this paper we deal with the following family of exponential maps $(f_λ:z\mapsto λ(e^z-1))_{λ\in [1,+\infty)}$. Denoting $d(λ)$ the hyperbolic dimension of $f_λ$. It is known that the function $λ\mapsto d(λ)$ is real analytic in $(1,+\infty)$, and that it is continuous in $[1,+\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0.
Let K be a compact subset of ${\mathbb R}^n$. We choose at random with uniform law a point at distance $ε$ of K and start a Brownian motion (BM) from this point. We study the probability that this BM hits K for the first time at a distance $\geq r$ from the starting point.