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Yûsuke Okuyama

Publications and source records attributed to Yûsuke Okuyama.

At least 19 recordsLinked to original sources
arXiv2026

Selberg and Brolin on value distribution of complex dynamics

The Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem for iterated preimages of a given non-exceptional value and Lyubich's periodic point version of it are foundational in the study of dynamics of rational functions of degree more than one on the complex projective line, and Drasin and the author studied a quantification of the former in a formalism of Nevanlinna theory, or more specifically, with the aid of Selberg's theorem from covering theory. In this paper, we point out that the argument in that previous study has already yielded a better quantification of the Brolin-Lyubich-Freire--Lopes--Mañé equidistribution theorem, and also point out that a similar argument also yields a quantification of Lyubich's theorem under an exponentwise version of the so called hypothesis H, which has recently been settled by Przytycki for unicritical polynomials. As an application, we sharpen the quantitative results on the value distribution of the (first) derivatives of iterated polynomials.

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arXiv2026

Equidistribution, multipliers, and moving targets in polynomial families

We study equidistribution of parameters for each of which a polynomial has a cycle with a prescribed moving multiplier in the full critically marked or unicritical polynomial families of a given degree $d>1$. For the full critically marked monic polynomial family $f=(f_λ)$, $λ\in\mathbb{C}^{d-1}$, of degree $d$, letting $p_{f,n}^*(λ,w)$ be the multiplier polynomial for cycles of formal exact period $n$ having multiplier $w$, we show that for any target sequence $w_n=d^na_n$ satisfying $n^{-1}\log|a_n|\to\ell\in[-\infty,+\infty)$, the averaged current of integration $ν(n)^{-1}[p_{f,n}^*(\cdot,w_n)=0]$ along the zero divisor of $p_{f,n}^*(\cdot,w_n)$ tends to the $(1,1)$-current $\mathrm{d}\mathrm{d}^c\max\{L_d,\log d+\ell\}$ in the sense of currents on $\mathbb{C}^{d-1}$ as $n\to\infty$, where $L_d$ is the Lyapunov exponent function and $ν(n)$ is the degree of the $n$th formal exact dynatomic polynomial associated to $f$. We also obtain the corresponding equidistribution theorem for the unicritical family $f_λ(z)=z^d+λ$, $λ\in\mathbb{C}$, of degree $d$. The proof combines thermodynamic formalism, the Eremenko--Levin multiplier bound, and a comparison principle for plurisubharmonic functions, in a partly quantitative manner. A key ingredient is the vanishing of the bifurcation measure on the boundary of every component of the interior of the connectedness locus for the full critically marked family, which we show using Kiwi's study of critical portraits and rational laminations.

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arXiv2026

Intrinsic semistable reduction loci for the iterations of non-archimedean quadratic rational functions and moduli-theoretic applications

We study the intrinsic semistable reduction loci for the iterations of quadratic rational functions over algebraically closed complete non-archimedean fields. Using a reduction-theoretic slope formula for the hyperbolic resultant function and the non-archimedean argument principle, we establish a precise eventual stationarity theorem when the residue characteristic is different from $2$. In the finite-order case, we define an intrinsic prehole parameter, and for quadratic rational functions with the same reduction order and boundary period, classify the intrinsic reduction data of all iterations at distinguished Fatou boundary points up to simultaneous projective equivalence. For antispecialization algebraic arcs degenerating to an order $p>1$ boundary point of the compactified quadratic moduli space over an algebraically closed field of characteristic $0$, we define an intrinsic antispecialization parameter using the fixed-point multiplier polynomial. We show that this parameter equals the prehole parameter of every Puiseux lift of an antispecialization arc degenerating to the boundary point and that the $p$th iteration specialization determines all iteration specializations. Over $\mathbb{C}$, the two intrinsic parameters coincide with DeMarco's parameter $τ^2$ after an appropriate interpretation. As a global consequence, we construct explicit blow-up models for the iterate compactification over arbitrary algebraically closed fields of characteristic $0$ and describe its boundary intrinsically. Over $\mathbb{C}$, this also recovers DeMarco's compactification.

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arXiv2026

Semistable reductions and minimalities of invariants for group scheme actions on projective schemes

Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\operatorname{SSRL}_x$ in the translation space $\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumption, also their non-emptiness. In the dynamical setting which has been studied by Szpiro--Tepper--Williams and Rumely, the coincidence result is already new in higher dimensions, and the non-emptiness result includes Rumely's $1$-dimensional result at least in the spherical complete case.

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arXiv2025

The intrinsic reductions and the intrinsic depths in non-archimedean dynamics

In this short paper, we aim at giving a more conceptual and simpler proof of Rumely's moduli theoretic characterization of type II minimal locus of the resultant function $\operatorname{ordRes}_ϕ$ on the Berkovich hyperbolic space for a rational function $ϕ$ on $\mathbb{P}^1$ defined over an algebraically closed and complete field that is equipped with a non-trivial and non-archimedean absolute value, and also aim at giving a much simpler and more natural proof of a degenerating limit theorem, in an improved form after DeMarco--Faber, for the family of the unique maximal entropy measures on $\mathbb{P}^1(\mathbb{C})$ associated to a meromorphic family of complex rational functions. We introduce the intrinsic reduction of a non-archimedean rational function $ϕ$ at each point in the Berkovich projective line and its directionwise intrinsic depths, which are suitable notions for the above aims and defined in terms of the tree and analytic structures of the Berkovich projective line. Then we establish two theorems in non-archimedean dynamics, both of which play key roles in the above aims.

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arXiv2024

Resultant measures and minimal resultant loci for non-archimedean polynomial dynamics

We compute the resultant measures for iterations $P^j$, $j\ge 1$, of a polynomial $P$ of degree $>1$ on the $n$-th level Trucco's trees $Γ_n$, $n\ge 0$, in the Berkovich projective line over a non-archimedean field and also determine their barycenters. As applications, we study the asymptotic of those barycenters as $n\to\infty$, and establish a uniform stationarity of Rumely's minimal resultant loci of $P^j$ or equivalently that of the potential semistable reduction loci of $P^j$ as $j\to\infty$. We also establish several equidistribution results for the resultant measures themselves as $n\to\infty$.

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arXiv2024

Parabolic bifurcation loci in the spaces of rational functions

We give a geometric description of the parabolic bifurcation locus in the space $\operatorname{Rat}_d$ of all rational functions on $\mathbb{P}^1$ of degree $d>1$, generalizing the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.

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arXiv2024

On a degenerating limit theorem of DeMarco--Faber

One of our aims is to complement the proof of DeMarco--Faber's degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco--Faber's more conceptual transfer principle between those dynamics.

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arXiv2023

A dynamical system over a non-archimedean field

This is an expository article, originally written in Japanese, on a dynamical system over a non-archimedean field. The main viewpoint is from complex and non-archimedean potential theories. After quickly introducing the Berkovich projective line, the dynamical moduli space as a scheme, and the various height functions on the space of rational functions and on the dynamical moduli space, we first survey our study of Rumely's new equivariants in non-archimedean dynamics and then survey our complex geometric and arithmetic studies of the dynamical moduli space from our joint works with Thomas Gauthier and Gabriel Vigny. The latter include a precise version of McMullen's finiteness theorem on formally exact multiplier spectra and an effective solution of Silverman's conjecture on a comparison between the moduli height and the critical height (qualitatively, the Silverman-Ingram theorem). The final topic is a degeneration of complex dynamics.

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arXiv2023

Equidistribution of the zeros of higher order derivatives in polynomial dynamics

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$ as $n\to+\infty$, when $f$ has no exceptional points in $\mathbb{C}$. This complements our former study on the zeros of $(f^n)^{(m)}-a$ for any value $a\in\mathbb{C}\setminus\{0\}$. The key in the proof is an approximation of the higher order derivatives of a solution of the Schröder or Abel functional equations for a meromorphic function on $\mathbb{C}$ with a locally uniform non-trivial error estimate.

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arXiv2020

Uniform perfectness of the Berkovich Julia sets in non-archimedean dynamics

We show that a rational function $f$ of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value has no potentially good reductions if and only if the Berkovich Julia set of $f$ is uniformly perfect. As an application, a uniform regularity of the boundary of each Berkovich Fatou component of $f$ is also established.

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arXiv2020

Entropy in uniformly quasiregular dynamics

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\ge 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f\colon M\to M$, the topological entropy $h(f)$ is $\log \mathrm{deg}( f )$. This proves Shub's entropy conjecture in this case.

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arXiv2019

Lehto--Virtanen-type and big Picard-type theorems for Berkovich analytic spaces

In non-archimedean setting, we establish a Lehto--Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk $\overline{\mathsf{D}}\setminus\{0\}$ in the Berkovich affine line to the Berkovich projective line $\mathsf{P}^1$ having an isolated essential singularity at the origin, and then establish a big Picard-type theorem for such an open subset $Ω$ in the Berkovich projective space $\mathsf{P}^N$ of any dimension $N$ that the family of all morphisms from $\overline{\mathsf{D}}\setminus\{0\}$ to $Ω$ is normal in a non-archimedean Montel's sense. As an application of the latter theorem, we see a big Brody-type hyperbolicity of the Berkovich harmonic Fatou set of an endomorphism of $\mathsf{P}^N$ of degree $>1$.

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arXiv2019

Geometric formulas on Rumely's weight function and crucial measure in non-archimedean dynamics

We introduce the $f$-crucial function $\operatorname{Crucial}_f$ associated to a rational function $f\in K(z)$ of degree $>1$ over an algebraically closed field $K$ of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely's (resultant) function $\operatorname{ordRes}_f$ in terms of the hyperbolic metric $ρ$ on the Berkovich upper half space $\mathsf{H}^1$ in the Berkovich projective line $\mathsf{P}=\mathsf{P}(K)$. We also obtain geometric formulas for Rumely's weight function $w_f$ and crucial measure $ν_f$ on $\mathsf{P}^1$ associated to $f$, as well as improvements of Rumely's principal results. As an application to dynamics, we obtain a quantitative equidistribution of the sequence $(ν_{f^n})_n$ of $f^n$-crucial measures towards the $f$-equilibrium (or canonical) measure $μ_f$ on $\mathsf{P}^1$.

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arXiv2019

Value distribution of derivatives in polynomial dynamics

For every $m\in\mathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $\mathbb{C}\setminus\{0\}$ under the $m$-th order derivatives of the iterates of a polynomials $f\in \mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $\mathbb{P}^1(\overline{k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb{C}^2$ has a given eigenvalue.

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arXiv2019

An a priori bound of endomorphisms of $\mathbb{C}\mathbb{P}^k$ and a remark on the Makienko conjecture in dimension one

Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and assume that for any cyclic Fatou component $W$ of $f$ having a period $p\in\mathbb{N}$, the equilibrium measure $μ_f$ has a positive charge on the boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform a priori bound of the dynamics of $f$, which in particular yields a Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We also point out that in the case of $k=1$, the statement of our assumption is related to both the impossibility for the Julia set of $f$ to be the boundary of lakes of Wada and the so called Makienko conjecture on the non-emptiness of the residual Julia set of $f$.

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arXiv2019

An a priori bound of rational functions on the Berkovich projective line

We establish a locally uniform a priori bound on the dynamics of a rational function $f$ of degree $>1$ on the Berkovich projective line over an algebraically closed field of any characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and deduce an equidistribution result for moving targets towards the equilibrium (or canonical) measure $μ_f$, under the no potentially good reductions condition. This partly answers a question posed by Favre and Rivera-Letelier.

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