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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., 18 Pages

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., 22 pages. To appear in Publ. RIMS

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 $��$ 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 $��$ 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 $>1$., 7 pages

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 H��non-type polynomial automorphism of $\mathbb{C}^2$ has a given eigenvalue.

Yuji Okuyama,* Kazuto Oya,* Shinji Matsunaga, Taro Kishi, Nakao Iwata Department of Psychiatry, Fujita Health University School of Medicine, Toyoake, Japan *These authors contributed equally to this work Abstract: This study aimed to perform a comprehensive meta-analysis of topiramate-augmen­tation therapy in patients with schizophrenia receiving antipsychotic agents. Data published up to June 20, 2016 were obtained from the PubMed, PsycINFO, and Cochrane Library databases. Twelve randomized controlled trials comparing topiramate to placebo or antipsychotic only were included (n=676 patients). The primary outcome was change in overall symptoms. Relative risk (RR) and standardized mean difference (SMD), along with 95% confidence intervals, were calculated using random effects model for each outcome. Topiramate-augmentation therapy was superior to the control for decreasing overall symptoms (SMD -0.55, 95% confidence interval -0.86 to -0.24; P=0.001; I2=55%, eight comparisons, n=380), positive symptoms (SMD -0.4), negative symptoms (SMD -0.47), and Positive and Negative Syndrome Scale general subscale scores (SMD -0.67). Furthermore, topiramate-augmentation therapy decreased weight (SMD -0.69) and body mass index (SMD -0.95) compared with the control. Topiramate was similar to the control with respect to discontinuation due to all causes (RR 1.19), inefficacy (RR 1.71), and adverse events (RR 1.09). Topiramate was associated with higher incidence of paresthesia (RR 2.67) and attention difficulty (RR 8.97) compared with the control. Our results seemed to suggest that topiramate-augmentation therapy improves the psychopathology of schizophrenia with good tolerability and has the additional advantage of weight maintenance. However, because there were some limitations (numbers of studies and patients included in the meta-analysis were small, some studies used completer analysis, Chinese studies were included in the meta-analysis, and studies that had a risk of bias were included in the meta-analysis) in this study, we cannot apply the results of this study in daily clinical practice. Keywords: schizophrenia, topiramate, efficacy, safety, systematic review, meta-analysis

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 $��_f$, under the no potentially good reductions condition. This partly answers a question posed by Favre and Rivera-Letelier., 10 pages. (v2) The complex counterpart is separated and the title is shortened

In the space $\mathbb{C}$ of the parameters $��$ of the unicritical polynomials family $f(��,z)=f_��(z)=z^d+��$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed measure) $T_f$ of $f$ as $n\to\infty$ on the averaged distributions of all parameters $��$ such that $f_��$ has a superattracting periodic point of period $n$ in $\mathbb{C}$, with a concrete error estimate for $C^2$-test functions on $\mathbb{P}^1$. In the proof, not only complex dynamics but also a standard argument from the Nevanlinna theory play key roles., 14 pages. (v2) A few words are corrected

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 $��_f$ 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 $f$., 6 pages. (v4) The complex part in arXiv:1805.07668v1 is marged with the title modification. We point out that in one dimension, the assumption in the main theorem is related to the impossibility for the Julia set set of $f$ to be the boundary of lakes of Wada and the Makienko conjecture, but those two problems are still open (v3) one reference is added (v2) more comments are added in Remark 2.2

For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a H��non map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any H��non map is accumulated by the bifurcation locus in the space of quadratic holomorphic endomorphisms of $\mathbb {P}^2$.

We look at degenerating meromorphic families of rational maps on $\mathbb{P}^1$ -- holomorphically parameterized by a punctured disk -- and we provide examples where the bifurcation current fails to have a bounded potential in a neighborhood of the puncture. This is in contrast to the recent result of Favre-Gauthier that we always have continuity across the puncture for families of polynomials; and it provides a counterexample to a conjecture posed by Favre in 2016. We explain why our construction fails for polynomial families and for families of rational maps defined over finite extensions of the rationals $\mathbb{Q}$., 13 pages

We establish a Mahler-type estimate of weighted Fekete sums on the Berkovich projective line over an algebraically closed field of possibly positive characteristic that is complete with respect to a non-trivial and possibly non-archimedean absolute value., 14 pages; (v2) Added Remark 3.1 and moved examples to the final section

We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in $\mathbb{C}\setminus\{0\}$ under the derivatives of the iterations of a polynomials $f\in\mathbb{C}[z]$ of degree more than one towards the $f$-equilibrium (or canonical) measure $��_f$ on $\mathbb{P}^1$. We also show that for every $C^2$ test function on $\mathbb{P}^1$, the convergence is exponentially fast up to a polar subset of exceptional values in $\mathbb{C}$. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is also established., 12 pages. (v3) corrected a few typos; (v2) Theorem 3 now focuses on a parameter space analog of Theorem 2 for the unicritical polynomials family

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 $��$ 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 $��_f$ 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 $(��_{f^n})_n$ of $f^n$-crucial measures towards the $f$-equilibrium (or canonical) measure $��_f$ on $\mathsf{P}^1$., 39 pages. (v4) Final version. (v3) The title is modified to be more informative. An effective estimate of the size of the $f$-crucial set is added. Some typos are corrected. (v2) An application to dynamics is added