Recent zbMATH articles in MSC 30Dhttps://zbmath.org/atom/cc/30D2021-05-28T16:06:00+00:00WerkzeugEntropy of transcendental entire functions.https://zbmath.org/1459.370322021-05-28T16:06:00+00:00"Benini, Anna Miriam"https://zbmath.org/authors/?q=ai:benini.anna-miriam"Fornæss, John Erik"https://zbmath.org/authors/?q=ai:fornass.john-erik"Peters, Han"https://zbmath.org/authors/?q=ai:peters.hanThe paper takes under consideration the topological entropy of transcendental entire maps. The results of [\textit{W. Bergweiler}, Conform. Geom. Dyn. 4, No. 2, 22--34 (2000; Zbl 0954.30012); \textit{J. P. R. Christensen} and \textit{P. Fischer}, Acta Math. Hung. 73, No. 3, 213--218 (1996; Zbl 0928.28005)] lead to the conclusion that the topological entropy of a transcendental function is always strictly positive. As transcendental entire maps have infinite topological degree, it could be expected that the topological entropy is also infinite. The main result of the paper is as follows.
Theorem. Let \(f\) be a transcendental entire function and let \(N\in\mathbb{N}.\) There exists a non-empty bounded open set \(V\subset\mathbb{C}\) so that \(V\subset f(V)\) and such that any point in \(V\) has at least \(N\) preimages in \(V\), counted with multiplicity.
It follows from this result that in the case of transcendental entire maps their topological entropy is, in fact, infinite. It should be mentioned that the same conclusion was reached independently in [\textit{M. Wendt}, Zufällige Juliamengen und invariante Maße mit maximaler Entropie. University of Kiel (PhD Thesis) (2005)].
Reviewer: Ewa Ciechanowicz (Szczecin)Uniqueness of meromorphic solutions sharing values with a meromorphic function to \(w(z + 1)w(z - 1) = H(z)w^m(z)\).https://zbmath.org/1459.390472021-05-28T16:06:00+00:00"Chen, BaoQin"https://zbmath.org/authors/?q=ai:chen.baoqin"Li, Sheng"https://zbmath.org/authors/?q=ai:li.shengSummary: For the nonlinear difference equations of the form \(w(z + 1)w(z - 1) = h(z)w^m(z),\) where \(h(z)\) is a nonzero rational function and \(m = \pm 2, \pm 1,0\), we show that its transcendental meromorphic solution is mainly determined by its zeros, 1-value points and poles except for some special cases. Examples for the sharpness of these results are given.Dynamics of a family of meromorphic functions with two essential singularities which are not omitted values.https://zbmath.org/1459.370332021-05-28T16:06:00+00:00"Domínguez, P."https://zbmath.org/authors/?q=ai:dominguez.patricia"Sienra, Guillermo"https://zbmath.org/authors/?q=ai:sienra.guillermo-j-f"Hernández, I."https://zbmath.org/authors/?q=ai:hernandez.isabel|hernandez.ivanSummary: In this article we investigate the dynamics of the family \(F_{\lambda ,c,\mu} (z)= \lambda e^{1/(z^2+c)} + \mu\), where \(\lambda\), \(c\in \mathbb{C} \setminus \{0\}\) and \(\mu \in\mathbb{C} \setminus \{\pm i\sqrt{c}\}\), with two essential singularities which are not omitted values. Choosing a slice of the space of parameters, we prove that for certain parameters \(\lambda\), \(c\) and \(\mu\), the Fatou set contains a completely invariant and multiply connected attracting domain, a parabolic domain and a Siegel disc. Moreover, we prove that the triple \((F_{\lambda,c,\mu},U,V)\) is a polynomial-like mapping of degree two for certain values of the parameters \(\lambda\) \(c\), \(\mu\), and some domains \(U\) and \(V\). Also, some examples of the Fatou and Julia sets for the polynomial-like mapping are given.Meromorphic functions that share four or three small functions with their difference operators.https://zbmath.org/1459.300032021-05-28T16:06:00+00:00"Liu, Huifang"https://zbmath.org/authors/?q=ai:liu.huifang"Mao, Zhiqiang"https://zbmath.org/authors/?q=ai:mao.zhiqiangSummary: In this paper, we prove that non-constant meromorphic functions of finite order and their difference operators are identical, if they share four small functions ``IM'', or share two small functions and \(\infty\) CM. Our results show that a conjecture posed by Chen-Yi in [\textit{Z.-X. Chen} and \textit{H.-X. Yi}, Result. Math. 63, No. 1--2, 557--565 (2013; Zbl 1267.30077)] is still valid for shared small functions, and improve some earlier results obtained by \textit{X.-M. Li} and \textit{H.-X. Yi} [Bull. Korean Math. Soc. 53, No. 4, 1213--1235 (2016; Zbl 1345.30035)], \textit{F. Lü} and \textit{W. Lü} [Comput. Methods Funct. Theory 17, No. 3, 395--403 (2017; Zbl 1405.30030)]. We also study the uniqueness of a meromorphic function partially sharing three small functions with their difference operators.Nevanlinna theory and algebraic values of certain meromorphic functions.https://zbmath.org/1459.111552021-05-28T16:06:00+00:00"Chalebgwa, Taboka Prince"https://zbmath.org/authors/?q=ai:chalebgwa.taboka-princeExtending results of [\textit{G. J. Boxall} and \textit{G. O. Jones}, Int. Math. Res. Not. 2015, No. 4, 1141--1158 (2015; Zbl 1388.11041); ibid. 2015, No. 22, 12251--12264 (2015; Zbl 1329.30015)] the author studies the number of points and the density of algebraic points of bounded degree and bounded absolute height \(H(z,f(z))\le H\) on graphs, restricted to compact subsets of \({\mathbb C}\), of entire functions of finite order and positive lower order. The bounds they obtain have the form \(C(\log H)^\eta\), with \(\eta\) an explicit constant depending on the order and lower order of the function. The author obtains similar bounds for meromorphic functions using the definition of the order arising from Nevanlinna's theory. Let \(f\) be a nonconstant meromorphic function of order \(\varrho\) and lower order \(\lambda\) with \(0<\lambda,\varrho<\infty\). Let \(r>0\). Suppose that \(f\) is analytic in a neighbourhood of the disc \(|z|\le 6r\). Then there are at most \(C(\log H)^\eta\) complex numbers \(z\) such that \(|z|\le r\), \([\mathbb{Q}(z,f(z)):\mathbb{Q}]\le d\) and \(H(z,f(z))\le H\). The exponent \(\eta\) is explicit, depending only on \(\varrho\), while \(C\) is effective and depends on \(f\) and \(d\). The author compares his results with the thesis of \textit{P. Villlemot} [Zero estimates and value distribution of meromorphic functions (Lemmes de zéros et distribution des valeurs des fonctions méromorphes) Grenoble: Université Grenoble Alpes (Diss.) (2016)]. The author also studies in detail a specific meromorphic function where his method can be applied (while Villemot's theorem does not apply), which is a function introduced by J.P-Serre related with elliptic integrals of the third kind [the reviewer, Nombres transcendants et groupes algébriques. (Transcendental numbers and algebraic groups). Complété par deux appendices de Daniel Bertrand et Jean-Pierre Serre. 2e éd. Paris: Société Mathématique de France (SMF) (1987; Zbl 0621.10022)]. This function is of the form
\[
F(z)=\frac{\sigma(z+u)}{\sigma(z)\sigma(u)}{\mathrm{e}}^{-\zeta(u)z},
\]
where \(\sigma\) is a Weierstrass sigma function attached to a lattice \(L\) and \(\zeta\) the associated Weierstrass zeta function
(see Chap.~20 of [\textit{D. Masser}, Auxiliary polynomials in number theory. Cambridge: Cambridge University Press (2016; Zbl 1354.11002)]). Transcendence results for the values of \(F\) are known only when the invariant \(g_2\) and \(g_3\) of the corresponding Weierstrass \(\wp\) function are supposed to be algebraic [the reviewer, Math. Rep. Acad. Sci., R. Soc. Can. 1, 111--114 (1979; Zbl 0402.10037)]. Without such an assumption, the author obtains an upper bound of the form \(C(\log H)^{12}\) for the number of algebraic points of height \(\le H\) and degree \(\le d\) in some discs.
Reviewer: Michel Waldschmidt (Paris)Orders and types of the Wright and Mittag-Leffler functions.https://zbmath.org/1459.300012021-05-28T16:06:00+00:00"Kilbas, A. A."https://zbmath.org/authors/?q=ai:kilbas.anatolii-aleksandrovich"Lipnevich, V. V."https://zbmath.org/authors/?q=ai:lipnevich.v-vSummary: An entire function, with coefficients involving products and quotients of a finite number of gamma functions, is considered. The order and the type of such a function, known as the generalized Wright function, are evaluated. Applications are given to evaluate the orders and types of the generalized Mittag-Leffler functions will even and odd parameters and of the generalized hypergeometric function. Special cases involving in particular Wright, Mittag-Leffler and the confluent hypergeometric Kummer functions are presented.Algebraic values of certain analytic functions defined by a canonical product.https://zbmath.org/1459.111572021-05-28T16:06:00+00:00"Chalebgwa, Taboka P."https://zbmath.org/authors/?q=ai:chalebgwa.taboka-princeThis paper deals with upper bounds for the number of algebraic values of bounded height and bounded degree on the graph of an analytic transcendental function. The author first gives a survey of the topic, including results by E.~Bombieri and J.~Pila, J.~Pila, D.W.~Masser, E.~Besson, A.~Surroca, G.~Boxall and G.~Jones. Next, he drops the condition that the points which are counted lie on a compact subset of \(\mathbb{C}\); he answers a question raised by C.~Miller on the density of such points on graphs of functions of order \(< 1\) defined by a canonical product. He considers an entire function of order \(\varrho\) with \(0<\varrho\le 1/2\) of the form \(f(z)=\prod_{n\ge 0} (1-z/z_n)\) and gives an upper bound of the form \(C(\log H)^\eta\) for the number of points \(z\) outside a sector \(-\phi\le {\mathrm {arg}}z\le \pi\) with \([\mathbb{Q}(z,f(z)):\mathbb{Q}]\le d\) and \(H(z,f(z))\le H\), where \(H\) is the absolute logarithmic height. The exponent \(\eta\) is explicit, depending only on \(\varrho\), while \(C\) is effective and depends on \(f\), \(\phi\) and \(d\) but not on \(H\). The proof is an adaptation of that of [\textit{G. Boxall} and \textit{G. Jones}, Int. Math. Res. Not. 2015, No. 22, 12251--12264 (2015; Zbl 1329.30015)] using an auxiliary polynomial constructed in [\textit{D. Masser}, J. Number Theory 131, No. 11, 2037--2046 (2011; Zbl 1267.11091)].
Reviewer: Michel Waldschmidt (Paris)Meromorphy of local zeta functions in smooth model cases.https://zbmath.org/1459.112202021-05-28T16:06:00+00:00"Kamimoto, Joe"https://zbmath.org/authors/?q=ai:kamimoto.joe"Nose, Toshihiro"https://zbmath.org/authors/?q=ai:nose.toshihiroSummary: It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general \(( C^\infty )\) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as \(u(x, y) x^a y^b +\) flat function, where \(u(0, 0) \neq 0\) and \(a, b\) are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when \(a < b\), local zeta functions can be meromorphically extended to the half-plane \(\operatorname{Re}(s) > - 1 / a\) and their poles on the half-plane are contained in the set \(\{- k / b : k \in \mathbb{N} \operatorname{with} k < b / a \} \).On the complex oscillation of meromorphic solutions of nonhomogeneous linear differential equations with meromorphic coefficients.https://zbmath.org/1459.341942021-05-28T16:06:00+00:00"Chen, Chuang-Xin"https://zbmath.org/authors/?q=ai:chen.chuangxin"Cui, Ning"https://zbmath.org/authors/?q=ai:cui.ning"Chen, Zong-Xuan"https://zbmath.org/authors/?q=ai:chen.zongxuanSummary: In this paper, we study the higher order differential equation \(f^{\left( k\right)}+Bf=H\), where \(B\) is a rational function, having a pole at \(\infty\) of order \(n>0\), and \(H\equiv0\) is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.Unicity of meromorphic solutions of the Pielou logistic equation.https://zbmath.org/1459.300022021-05-28T16:06:00+00:00"Li, Sheng"https://zbmath.org/authors/?q=ai:li.sheng"Chen, Baoqin"https://zbmath.org/authors/?q=ai:chen.baoqinSummary: This paper mainly considers the unicity of meromorphic solutions of the Pielou logistic equation \(y\left( z + 1\right)=\left( \left( R \left( z\right) y \left( z\right)\right) / \left( Q \left( z\right) + P \left( z\right) y \left( z\right)\right)\right)\), where \(P\left( z\right)\), \(Q\left( z\right)\), and \(R\left( z\right)\) are nonzero polynomials. It shows that the finite order transcendental meromorphic solution of the Pielou logistic equation is mainly determined by its poles and 1-value points. Examples are given for the sharpness of our result.A note on entire functions sharing a finite set with their difference operators.https://zbmath.org/1459.300042021-05-28T16:06:00+00:00"Qi, Jianming"https://zbmath.org/authors/?q=ai:qi.jianming"Wang, Yanfeng"https://zbmath.org/authors/?q=ai:wang.yanfeng"Gu, Yongyi"https://zbmath.org/authors/?q=ai:gu.yongyiSummary: In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of \textit{X.-M. Li} [Comput. Methods Funct. Theory 12, No. 1, 307--328 (2012; Zbl 1260.30017)] and partially answers \textit{K. Liu}'s [``Meromorphic functions sharing a set with applications to difference equation'', J. Math. Anal. Appl. 359, 384--393 (2009; \url{doi:10.1016/j.jmaa.2009.05.061})] question.Anderson localization for long-range operators with singular potentials.https://zbmath.org/1459.810432021-05-28T16:06:00+00:00"Jian, Wenwen"https://zbmath.org/authors/?q=ai:jian.wenwen"Shi, Jia"https://zbmath.org/authors/?q=ai:shi.jia"Yuan, Xiaoping"https://zbmath.org/authors/?q=ai:yuan.xiaopingSummary: In this paper, we use the Cartan estimate for meromorphic functions to prove Anderson localization for a class of long-range operators with singular potentials.
{\copyright 2021 American Institute of Physics}Algebraic differential independence regarding the Riemann \(\boldsymbol{\zeta}\)-function and the Euler \(\boldsymbol{\Gamma}\)-function.https://zbmath.org/1459.111672021-05-28T16:06:00+00:00"Han, Qi"https://zbmath.org/authors/?q=ai:han.qi.1"Liu, Jingbo"https://zbmath.org/authors/?q=ai:liu.jingboThe Riemann zeta function \(\zeta\) and the Euler gamma function \(\Gamma\) are the usual ones. The main result of this paper is that \(\zeta\) cannot be a solution to any nontrivial algebraic differential equation whose coefficients are polynomials in \(\Gamma\), \(\Gamma^{(n)}\), \(\Gamma^{(ln)}\), for positive integers \(l\) and \(n\). Formally stated:
Theorem. Let \(l, m, n \geq 0\) be integers. Assume \(P(u_0, u_1, \dots, u_m; v_0, v_1, v_2)\) is a polynomial of \(m+4\) variables with polynomial coefficients in \(z \in \mathbb C\) such that
\[P(\zeta, \zeta', \dots, \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(ln)})(z) \equiv 0 \]
for \(z\in \mathbb C\). Then, necessarily the polynomial \(P\) must be identical to zero.
Reviewer: Stelian Mihalas (Timişoara)Uniqueness of meromorphic solutions of the difference equation \(R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z)\).https://zbmath.org/1459.390482021-05-28T16:06:00+00:00"Li, Sheng"https://zbmath.org/authors/?q=ai:li.sheng"Chen, BaoQin"https://zbmath.org/authors/?q=ai:chen.baoqinSummary: This paper mainly concerns the uniqueness of meromorphic solutions of first order linear difference equations of the form \[ R_{1}(z)f(z+1)+R_{2}(z)f(z)=R_{3}(z), \eqno{(*)}\] where \(R_{1}(z)\not \equiv 0\), \(R_{2}(z)\), \(R_{3}(z)\) are rational functions. Our results indicate that the finite order transcendental meromorphic solution of equation \((*)\) is mainly determined by its zeros and poles except for some special cases. Examples for the sharpness of our results are also given.