Recent zbMATH articles in MSC 37Fhttps://www.zbmath.org/atom/cc/37F2021-03-30T15:24:00+00:00WerkzeugTaylor coefficients of the conformal map for the exterior of the reciprocal of the multibrot set.https://www.zbmath.org/1455.370432021-03-30T15:24:00+00:00"Shimauchi, Hirokazu"https://www.zbmath.org/authors/?q=ai:shimauchi.hirokazuSummary: In this paper we investigate normalized conformal mappings of the exterior of the reciprocal of the Multibrot set and analyze the growth of the denominator of the coefficients. Our inequality improves Ewing and Schober's result which was presented in [\textit{J. H. Ewing} and \textit{G. Schober}, J. Math. Anal. Appl. 170, No. 1, 104--114 (1992; Zbl 0766.30012)]. We use the coefficient formula of the author [in: Topics in finite or infinite dimensional complex analysis. Proceedings of the 19th international conference on finite or infinite dimensional complex analysis and applications (ICFIDCAA), Hiroshima, Japan, December 11--15, 2011. Sendai: Tohoku University Press. 237--248 (2013; Zbl 1333.37055)]. The straightforward adaptation of the proof in this paper slightly improves the main theorem of the author [Osaka J. Math. 52, No. 3, 737--746 (2015; Zbl 1352.37142)].On interpreting Patterson-Sullivan measures of geometrically finite groups as Hausdorff and packing measures.https://www.zbmath.org/1455.370342021-03-30T15:24:00+00:00"Simmons, David"https://www.zbmath.org/authors/?q=ai:simmons.davidSummary: We provide a new proof of a theorem whose proof was sketched by \textit{D. Sullivan} [Acta Math. 149, 215--237 (1982; Zbl 0517.58028)], namely that if the Poincaré exponent of a geometrically finite Kleinian group \(G\) is strictly between its minimal and maximal cusp ranks, then the Patterson-Sullivan measure of \(G\) is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of \textit{B. O. Stratmann} [in: Non-Euclidean geometries. János Bolyai memorial volume. Papers from the international conference on hyperbolic geometry, Budapest, Hungary, July 6--12, 2002. New York, NY: Springer. 227--247 (2006; Zbl 1104.37020)].Harmonic measure: algorithms and applications.https://www.zbmath.org/1455.300152021-03-30T15:24:00+00:00"Bishop, Christopher J."https://www.zbmath.org/authors/?q=ai:bishop.christopher-jThis survey article discusses several results in the field of planar harmonic measure, starting from \textit{N. G. Makarov}'s results [Proc. Lond. Math. Soc. (3) 51, 369--384 (1985; Zbl 0573.30029)] to recent applications involving 4-manifolds, dessins d'enfants and transcendental dynamics. Various areas from analysis, topology and algebra that are influenced by harmonic measure are illustrated.
For the entire collection see [Zbl 1437.00045].
Reviewer: Marius Ghergu (Dublin)Thermodynamic formalism and integral means spectrum of logarithmic tracts for transcendental entire functions.https://www.zbmath.org/1455.300192021-03-30T15:24:00+00:00"Mayer, Volker"https://www.zbmath.org/authors/?q=ai:mayer.volker"Urbański, Mariusz"https://www.zbmath.org/authors/?q=ai:urbanski.mariuszIn this very interesting pioneering paper the authors develop an entirely new approach to the thermodynamic formalism for entire functions with bounded singular sets. This approach covers all entire functions for which the thermodynamic formalism has been so far established and goes far beyond. They show that there is a strong relation between the transfer operator, on whose properties the thermodynamic formalisms relay, and the integral means spectrum for logarithmic tracts. They derive from this that the negative spectrum property implies good behavior of this operator. The negative spectrum property turns out to be a very general condition which holds as soon as the tracts have some sufficiently nice geometry, for example, for quasidisks, John, or Hölder tracts. In these cases the authors get a good control of the corresponding transfer operators, leading to the full thermodynamic formalism along with its applications such as exponential decay of correlations, a central limit theorem, and a Bowen's formula for the Hausdorff dimension of radial Julia sets.
In particular, they show that the thermodynamic formalism holds for every hyperbolic function from the Eremenko-Lyubich analytic family of the Speicer class \(\mathcal{S}\) (the class \(S\) consists of entire functions with finite singular set), provided this family contains at least one function with Hölder tracts. The latter is, for example, the case if the Eremenko-Lyubich analytic family \(\mathcal{M}_g \) is associated with the Poincaré function of a polynomial \(g\in \mathcal{S}\) having a connected Julia set.
Reviewer: Olga M. Katkova (Boston)Analytic deformations of pencils and integrable one-forms having a first integral.https://www.zbmath.org/1455.370442021-03-30T15:24:00+00:00"Scárdua, Bruno"https://www.zbmath.org/authors/?q=ai:scardua.bruno|scardua.bruno-c-azevedoThe paper addresses the problem of persistence of a first integral for a holomorphic integrable one-form in a neighborhood of a singular point.
The author considers integrable analytic deformations of codimension-one holomorphic foliations. Let \(\Omega(x,t)\) be a holomorphic one-form defined in a connected product neighborhood \(U \times D\subset \mathbb{C}^n\times\mathbb{C}\), of the origin of \(\mathbb{C}^{n+1}\), where \(D\subset\mathbb{C}\) is a disc centered at the origin, such that \(\Omega^0:=\Omega(x,0)\) admits a first integral of holomorphic or meromorphic type. For each parameter \(t\in D\), \(\Omega^t\) denotes the restriction \(\Omega^t:=\Omega(x,t)\). The author considers the case where each \(\Omega^t\) is integrable. The family of codimension-one foliations \(\{\mathcal{F}_t:\Omega^t=0~\text{in}~U\}_{t\in(\mathbb{C},0)}\) can be seen as an analytic deformation in \(U\) of the foliation \(\mathcal{F}_0\) which, by hypothesis, admits a holomorphic (or meromorphic) first integral \(f\).
In the first part of the paper, the author studies the case where the foliation \(\mathcal{F}_0\) has a holomorphic first integral \(f\) at the origin in \(\mathbb{C}^n\) with \(n\ge 3\), under the assumption that the germ \(f\) is irreducible and that its typical fiber is simply-connected. In Theorem 1.1, the author first proves that if the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(2\), then the germ of the foliation \(\mathcal{F}_t\) also has a holomorphic first integral. Then, for the general case, that is when the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(1\), the author obtains a two-dimensional normal form for the foliation \(\mathcal{F}_t\).
The second part of the paper deals with analytic deformations \(\{\mathcal{F}_t\}_{t\in(\mathbb{C},0)}\) of a local pencil \(\mathcal{F}_0:\frac{f}{g}=\hbox{constant}\) for \(f,g\) belonging to the ring \(\mathcal{O}_n\) of germs of holomorphic functions at the origin of \(\mathbb{C}^n\). In dimension \(n=2\), the author considers \(f=x\) and \(g=y\) and assumes that the axes are invariant for each foliation \(\mathcal{F}_t\).
In higher dimension \(n\ge 3\), the author works under some generic geometric conditions on the germs \(f\) and \(g\). In both cases, the author proves the following (Theorems 1.2 and 1.3):
1) For an analytic deformation there is a multiform formal first integral of type \(\widehat F = \frac{f^{1+\widehat\lambda(t)}}{g^{1+\widehat\mu(t)}} e^{\widehat H(x,y,t)}\) with \(\widehat\lambda(t)\) and \(\widehat\mu(t)\) formal series and some properties on \(\widehat H(x,y,t)\);
2) For an integrable deformation there is a meromorphic first integration of the form \(M = \frac{f}{g} e^{P(t)+H(x,y,t)}\) with \(P(t)\) holomorphic and some additional properties on \(H(x,y,t)\).
Reviewer: Jasmin Raissy (Toulouse)Identifying logarithmic tracts.https://www.zbmath.org/1455.300202021-03-30T15:24:00+00:00"Waterman, James"https://www.zbmath.org/authors/?q=ai:waterman.jamesLet \(D\) be a direct tract of a holomorphic function \(f\). It is proved, if the boundary of \(D\) is an unbounded simple
curve, then \(D\) is a logarithmic tract, i.e., the restriction \(f:D\rightarrow\{z\in \mathbb{C}: |z|>R\}\) is a universal covering, where \(R\) is the boundary value of the direct tract. With the additional assumption that there are no asymptotic paths in a logarithmic tract \(D\) with finite asymptotic values, then the following converse to this assertion is true.
Theorem 1.2. Let \(D\) be a logarithmic tract containing no asymptotic paths
with finite asymptotic values. Then \(D\) is bounded by a single unbounded curve.
Further, if \(D\) is a logarithmic tract with boundary value \(R\), then for all \(R'>R\),
\(\{z\in D:|f(z)|>R'\}\) is a logarithmic tract bounded by a simple curve.
As an application of these results, it is shown that an example of a function with infinitely many direct singularities,
but no logarithmic singularity over any finite value, is in the Eremenko-Lyubich class.
Reviewer: Konstantin Malyutin (Kursk)A note on the growth of solutions of second-order complex linear differential equations.https://www.zbmath.org/1455.340912021-03-30T15:24:00+00:00"Qiao, Jianyong"https://www.zbmath.org/authors/?q=ai:qiao.jianyong"Zhang, Qi"https://www.zbmath.org/authors/?q=ai:zhang.qi|zhang.qi-shuhuason|zhang.qi.4|zhang.qi.2|zhang.qi.1"Long, Jianren"https://www.zbmath.org/authors/?q=ai:long.jianren"Li, Yezhou"https://www.zbmath.org/authors/?q=ai:li.yezhouLet \(g\) be an entire function. Then the order of growth of \(g\) is defined by
\[
\rho (g)=\limsup_{r\rightarrow +\infty} \frac{\log ^{+}T(r,g)}{\log r}=\limsup_{r\rightarrow +\infty}\frac{\log \log^{+}M(r,g)}{\log r},
\]
where \(T(r,g)\) is the Nevanlinna characteristic function and \(M(r,g)=\max_{|z|=r}|g(z)|\).
Let \(\alpha \), \(\beta \) be two constants with \(0\leq \alpha<\beta \leq 2\pi\). Set
\begin{gather*}
\Omega (\alpha,\beta)=\left\{ z\in\mathbb{C}:\alpha <\arg z<\beta \right\} , \\
\overline{\Omega}(\alpha,\beta)=\left\{ z\in\mathbb{C}:\alpha \leq \arg z\leq \beta \right\} , \\
\Omega \left( \alpha ,\beta ,r\right) =\left\{ z\in\mathbb{C}:\alpha <\arg z<\beta \right\} \cap \left\{ z\in\mathbb{C}:\left\vert z\right\vert <r\right\}.
\end{gather*}
Let \(g\) be an analytic function in \(\overline{\Omega}\left( \alpha ,\beta\right) \). Then the growth order \(\rho_{\alpha ,\beta}(g)\) of \(g\) in \(\Omega (\alpha,\beta)\) is defined as
\[
\rho_{\alpha ,\beta}(g)=\limsup_{r\rightarrow +\infty} \frac{\log ^{+}\log ^{+}M(r,\Omega (\alpha,\beta),g)}{\log r},
\]
where \(M(r,\Omega (\alpha,\beta),g)=\sup_{\alpha \leq \arg z\leq \beta}\left\vert g\left( re^{i\theta}\right) \right\vert\). Moreover the radial order \(\rho_{\theta}(g)\) of \(g\) is denoted by
\[
\rho_{\theta}(g)=\lim_{\varepsilon \rightarrow 0^{+}}\limsup_{r\rightarrow +\infty} \frac{\log ^{+}\log^{+}M(r,\Omega \left( \alpha -\varepsilon ,\beta +\varepsilon \right) ,g)}{\log r}.
\]
In this paper under review, the authors study the growth of solutions of the second-order linear differential equation
\[
f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0,
\tag{1}
\]
where \(A(z)\) and \(B(z)\) are entire functions.
Firstly, the authors assumes that the coefficient \(A(z)\) of \((1)\) is a solution of another equation. Secondly, they consider the lower bound estimate of the linear measure of the angular domain \(I(f)=\left\{ \theta \in \left[ 0,2\pi \right] :\rho_{\theta}(f)=\infty \right\} \) of every non-trivial solution \(f\) of \((1)\).
The results obtained improve and extend those of \textit{J. R. Long} [``On the radial distribution of Julia set of solutions of \(f^{\prime\prime}+A(z)f^{\prime}+B(z)f=0\)'', J. Comput. Anal. Appl. 24, No. 4, 675--691 (2018); \textit{Z.-G. Huang} and \textit{J. Wang}, J. Math. Anal. Appl. 431, No. 2, 988--999 (2015; Zbl 1320.30057)] and [\textit{Z. Zhou} et al., J. Guizhou Norm. Univ., Nat. Sci. 31, No. 2, 50--53, 111 (2013; Zbl 1289.30202)].
Some examples are given to illustrate the results obtained.
Reviewer: Benharrat Belaidi (Mostaganem)On generalized Lattès maps.https://www.zbmath.org/1455.300252021-03-30T15:24:00+00:00"Pakovich, Fedor"https://www.zbmath.org/authors/?q=ai:pakovich.fedorSummary: We introduce a class of rational functions \(A:\mathbb{CP}^1\rightarrow\mathbb{CP}^1\) which can be considered as a natural extension of the class of Lattès maps, and establish basic properties of functions from this class.A characterization of multiplicity-preserving global bifurcations of complex polynomial vector fields.https://www.zbmath.org/1455.370422021-03-30T15:24:00+00:00"Dias, Kealey"https://www.zbmath.org/authors/?q=ai:dias.kealeySummary: For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree \(d\), it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized.Braid equivalence in the Hénon family. I.https://www.zbmath.org/1455.370412021-03-30T15:24:00+00:00"de Carvalho, A."https://www.zbmath.org/authors/?q=ai:de-carvalho.andre-f|salles-de-carvalho.andre|nolasco-de-carvalho.alexandre|de-carvalho.alexandre-luis-trovon|de-carvalho.a-a|ponce-de-leon-ferreira-de-carvalho.andre-carlos|de-carvalho.alcides"Hall, T."https://www.zbmath.org/authors/?q=ai:hall.t-j|hall.tony|hall.tom-e|hall.tyson-s|hall.tracy|hall.toby|hall.t-m|hall.thomas-eric"Hazard, P."https://www.zbmath.org/authors/?q=ai:hazard.p-e|hazard.peterThe dynamics of the real quadratic family \(f_{a}(x)\,=\,a-x^{2}\) is well understood (see [\textit{M. Lyubich}, Notices Am. Math. Soc. 47, No. 9, 1042--1052 (2000; Zbl 1040.37032)]). On the other hand the understanding of the two-dimensional Hénon family \(F_{a,\,b}(x,\,y)\,=\,(f_{a}(x)-by,\,x)\) (which for \(b\,=\,0\) degenerates to \(f_{a}(x)\)) is still rather rudimentary.
This article is concerned with periodic orbits in the Hénon familly in the parameter regions close to degeneration, and exploits both similarities and differences between the quadratic family and
Hénon family.
In the paper, the authors describe two mechanisms (Constructions 3.1 and 4.6) which lead to a coalescence on the
level of unimodal permutations (the unimodal permutations are defined in Definition 2.8).
The paper is rather technical and the used terminology cannot be totally reproduced here.
A geometric
braid on \(n\) strands is a diagram with \(n\) arcs (strands) connecting two
ordered sets of \(n\) points lined up vertically, so that only double intersections are
allowed and at each of them it is specified which strand goes above and which goes
below. To each geometric braid is associated a braid type, and
braid types determine geometric braids up to conjugacy.
For a \(p\in \mathbb{N}\) with \(U_{p}\) it is denoted the set of unimodal
permutations on \(p\) symbols.
For any \(u\in U_{p}\) there is unique unimodal braid \(\beta\) which induces \(u\). The set of unimodal braids on \(p\) strands is denoted by \(UB_{p}\).
Theorem A (Theorem 3.3 in the paper). Let \( p \in \mathbb{N}\). Let \(\beta \in UB_{p}\) be cyclic. Assume \(\beta \) is reconnectable at the non-dynamical preimage. Let \(\beta _{-}\) and \(\beta _{+}\) denote the braids from Construction 3.1.
1. If the non-dynamical preimage lies to the left of \(m\), then \(\beta _{+}\,\sim\,\beta _{-}\).
2. If the non-dynamical preimage lies to the right of \(m\), then \(\beta _{+}\,\sim_{r}\,\beta _{-}\).
Theorem B (Theorem 4.8 in the paper). Given \((u_{-}, \,u_{+})\), satisfying Properties (1)--(4) (page 97 in the paper). Let \(\beta _{-}\) and \(\beta _{+}\) denote the pair of braids produced
from Construction 4.6.
1. If \(\beta _{-}\,\sim\,\beta _{+}\), then \(\beta _{-}^{1}\,\sim\,\beta _{+}^{1}\).
2. If \(\beta _{-}\,\sim _{r}\,\beta _{+}\), then \(\beta _{-}^{1}\,\sim _{r}\,\beta _{+}^{1}\).
Moreover, the corresponding unimodal permutations \((u_{-}^{1}\,\sim _{r}\,u_{+}^{1})\) also satisfy the Properties (1)--(4).
The two above results are applied to equivalences realised in the Hénon family.
The paper ends with two questions.
Question A. Let \(a_{-}, \,\,a_{+}\in [-1/4,\,\, 2]\) be such that \(f_{a_{-}}\) and
\(f_{a_{+}}\) have critical orbits \(c_{-}\) and \(c_{+}\)
of types \(u_{-}\) and \(u_{+}\) respectively. Let \(C_{-}\) and \(C_{+}\) denote the corresponding periodic orbits
for \(F_{a_{-},\,0}\) and \(F_{a_{+},\,0}\) respectively. Does there exist a braid equivalence in the family \(F_{a,\,b}\)
connecting \((C_{-},\, F_{a_{-},\,0})\) and \((C_{+},\, F_{a_{+},\,0})\)?
Question B. For each positive integer \(i\), let \(a^{i}_{-} ,\,\,a^{i}_{+}\in [-1/4,\,\, 2]\) be parameters such that \(f_{a^{i}_{-}}\) and
\(f_{a^{i}_{+}}\) have critical orbits \(c^{i}_{-}\) and \(c^{i}_{+}\)
of types \(u^{i}_{-}\) and \(u^{i}_{+}\) respectively.
Let \(C^{i}_{-}\) and \(C^{i}_{+}\) denote the corresponding periodic orbits
for \(F_{a^{i}_{-},\,0}\) and \(F_{a^{i}_{+},\,0}\) respectively. Does there exist,
for each \(i\), a braid equivalence in the family \(F_{a,\,b}\)
connecting \((C^{i}_{-},\, F_{a^{i}{-},\,0})\) and \((C^{i}_{+},\, F_{a^{i}{+},\,0})\)?
Are the paths \(\gamma ^{i}\) in the \((a,\, b)\)-plane which realise these braid equivalences pairwise disjoint?
For the entire collection see [Zbl 1432.37002].
Reviewer: Dimitrios Varsos (Athína)