arXiv subjects
Publications and source records attributed to Reimi Irokawa.
For an analytic family $\{f_t\}_{t\in\mathbb{D}^*}$ on the unit punctured disk that meromorphically degenerates at the origin, we show that its limiting measure on an snc model is given by the push forward of the canonical measure attached to the non-archimedean rational function naturally induced from the family, which is a generalization of the results by DeMarco-Faber and Okuyama.
We study degenerating families of hyperbolic dynamics over complex K3 surfaces by means of the theory of hybrid spaces by Boucksom, Favre, and Jonsson. For an analytic family of hyperbolic automorphisms $\{f_t: X_t\to X_t\}_{t\in\mathbb{D}^*}$ over K3 surfaces $X_t$ that is possibly meromorphically degenerating at the origin, we consider the family of invariant measures $\{η_t\}$ on $X_t$ constructed by Cantat. The family $f_t$ induces a hyperbolic automorphism $f_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}:X_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}\to X_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}$ over the induced non-archimedean K3 surface, where we also have a measure $η_0$ by Filip. Our main theorem states the weak convergence of $\{η_t\}$ to $η_0$ as $t\to0$ over the induced so-called hybrid space.
For studying the meromorphic degeneration of complex dynamics, the theory of hybrid spaces, introduced by Boucksom, Favre and Jonsson, is known to be a strong tool. In this paper, we apply this theory to the dynamics of Hénon maps. For a family of Hénon maps $\{H_t\}_{t\in\mathbb{D}^*}$ that is parametrized by a unit punctured disk and meromorphically degenerates at the origin, we show that as $t\to 0$, the family of the invariant measures $\{μ_t\}$ "weakly converges" to a measure on the Berkovich affine plane associated to the non-archimedean Hénon map determined by the family $\{H_t\}_t$. We also calculate the limit of their Lyapunov exponents.
For a cubic rational function with coefficients in a non-archimedean field $K$ whose residue characteristic is $0$ or greater than $3$, there are $2$ possibilities for the shape of its Berkovich ramification locus, considered as an endomorphism of the Berkovich projective line: one is the connected hull of all the critical points, and the other is consisting of $2$ disjoint segments. In this paper, we list up all the possible forms of cubic rational functions and calculate their ramification loci.
Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $\mathbb{P}^{1,an}$ over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we study more about the activity locus such as its relation to boundedness locus, i.e., the Mandelbrot set, and to the normality of the sequence of the forward orbit.