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We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of
$F.$
As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.
We consider the family
$\mathrm {MC}_d$
of monic centered polynomials of one complex variable with degree
$d \geq 2$
, and study the map
$\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$
which maps each
$f \in \mathrm {MC}_d$
to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$
for every
$\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$
except when the fiber
$\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$
contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math.322 (2017), 132–185] which gave a rather long algorithm with some induction processes.
Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial
$f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$
we show that the Mahler measure of the iterates
$f^n$
grows geometrically fast with the degree
$d^n,$
and find the exact base of that exponential growth. This base is expressed via an integral of
$\log ^+|z|$
with respect to the invariant measure of the Julia set for the polynomial
$f.$
Moreover, we give sharp estimates for such an integral when the Julia set is connected.
We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by
\begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*}
Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit
\begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*}
of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.
In his 1985 paper, Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper, we relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.
The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.
We consider the McMullen maps
$f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$
with
$\unicode{x3bb} \in \mathbb {C}^{*}$
and
$n \geq 3$
. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between
$1$
and
$2$
.
In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component could be bounded even if the orbit of the component contains a sequence of annuli whose moduli tend to infinity, and this cannot happen when the maximal modulus of the meromorphic function is uniformly large enough. In this way we extend certain related results for entire functions to meromorphic functions with infinitely many poles.
The set of points that escape to infinity under iteration of a cosine map, that is, of the form
$C_{a,\,b} \colon z \mapsto ae^z+be^{-z}$
for
$a,\,b\in \mathbb{C}^\ast$
, consists of a collection of injective curves, called dynamic rays. If a critical value of
$C_{a,\,b}$
escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map
$z\mapsto \cosh(z)$
. We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of
$z\mapsto \cosh(z)$
, and conclude that no two of its dynamic rays land together.
We consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree
$2$
on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.
A parameter
$c_{0}\in {\mathbb {C}}$
in the family of quadratic polynomials
$f_{c}(z)=z^{2}+c$
is a critical point of a period n multiplier if the map
$f_{c_{0}}$
has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative at
$c=c_{0}$
. We study the accumulation set
${\mathcal X}$
of the critical points of the multipliers as
$n\to \infty $
. This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper, we prove that the accumulation set
${\mathcal X}$
is bounded, connected, and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set
${\mathcal X}$
and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends c to the Hausdorff dimension of
$f_{c}$
does not have critical points outside of the accumulation set
${\mathcal X}$
.
Since 1984, many authors have studied the dynamics of maps of the form
$\mathcal{E}_a(z) = e^z - a$
, with
$a > 1$
. It is now well-known that the Julia set of such a map has an intricate topological structure known as a Cantor bouquet, and much is known about the dynamical properties of these functions.
It is rather surprising that many of the interesting dynamical properties of the maps
$\mathcal{E}_a$
actually arise from their elementary function theoretic structure, rather than as a result of analyticity. We show this by studying a large class of continuous
$\mathbb{R}^2$
maps, which, in general, are not even quasiregular, but are somehow analogous to
$\mathcal{E}_a$
. We define analogues of the Fatou and the Julia set and we prove that this class has very similar dynamical properties to those of
$\mathcal{E}_a$
, including the Cantor bouquet structure, which is closely related to several topological properties of the endpoints of the Julia set.
In this paper, we study the dynamics of the Newton maps for arbitrary polynomials. Let p be an arbitrary polynomial with at least three distinct roots, and f be its Newton map. It is shown that the boundary
$\partial B$
of any immediate root basin B of f is locally connected. Moreover,
$\partial B$
is a Jordan curve if and only if
$\mathrm {deg}(f|_B)=2$
. This implies that the boundaries of all components of root basins, for the Newton maps for all polynomials, from the viewpoint of topology, are tame.
We prove that the set of all endpoints of the Julia set of
$f(z)=\exp\!(z)-1$
which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space
$\mathfrak E$
. As a corollary, we show that the set of all points
$z\in \mathbb C$
whose orbits either escape to
$\infty$
or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.
Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.
We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda $ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics but have also been key in the recent deterministic approximation scheme for all $|\lambda |\neq 1$ by Liu, Sinclair and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle and on the tantalising question of what happens around $\lambda =1$, where, on one hand, the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability and, on the other hand, the presence of Lee–Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda =1$ and, more generally, on the entire unit circle. For an integer $\Delta \geq 3$ and edge interaction parameter $b\in (0,1)$, we show $\mathsf {\#P}$-hardness for approximating the partition function on graphs of maximum degree $\Delta $ on the arc of the unit circle where the Lee–Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda |\neq 1$ or when $\lambda $ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee–Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.
A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc.374(9) (2021), 6145–6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.
In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.
We construct automorphisms of
${\mathbb C}^2$
, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form
$F(z,w)=(g(z,w),z)$
with
$g(z,w):{\mathbb C}^2\rightarrow {\mathbb C}$
holomorphic.
The Julia set of the exponential family
$E_{\kappa }:z\mapsto \kappa e^z$
,
$\kappa>0$
was shown to be the entire complex plane when
$\kappa>1/e$
essentially by Misiurewicz. Later, Devaney and Krych showed that for
$0<\kappa \leq 1/e$
the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of
$\mathbb {R}^3$
, generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in
$\mathbb {R}^3$
and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.