In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d(\mathbb P^2)$ and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.
A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and D{í}az in the 90s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper, we introduce analogous notions in a measurable point of view. We define an almost blender as a hyperbolic basic set for which a prevalent p…
Let $f$ be a Hénon-Sibony map (regular polynomial automorphism) of $\mathbb{C}^k$ and let $μ$ be the equilibrium measure of $f$. In this paper we prove that $μ$ is exponentially mixing for plurisubharmonic test functions.
We prove an equidistribution theorem of positive closed currents for a certain class of birational maps $f_+:\mathbb{P}^k\to\mathbb{P}^k$ of algebraic degree $d\geq 2$ satisfying $\bigcup_{n\geq 0}f_-^n(I^+)\cap \bigcup_{n\geq 0}f_+^n(I^-)=\emptyset$, where $f_-$ is the inverse of $f_+$ and $I^\pm$ are the sets of indeterminacy for $f_\pm$, respectively.
Let $(φ_t)$, $(ϕ_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb D\subset \mathbb C$. Let $f:\mathbb D \to \mathbb D$ be a homeomorphism. We prove that, if $f \circ ϕ_t=φ_t \circ f$ for all $t\geq 0$, then $f$ extends to a homeomorphism of $\bar{\mathbb D}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, $f$ extends to a homeomorphism of $\bar{\mathbb D}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disc.
A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $ϕ$ of the unit disk $\mathbb{D}$ normalized by $ϕ(0) = 0$ and $ϕ'(0) > 0$ can be obtained as the reachable set of the Loewner - Kufarev control system $$ \frac{\mathrm{d} w_t}{\mathrm{d} t}=G_t\circ w_t,\quad t\geqslant0,\qquad w_0=\mathsf{id}_{\mathbb{D}}, $$ where the control functions $t\mapsto G_t\in\mathsf{Hol}(\mathbb{D},\mathbb{C})$ form a certain convex cone. Here we extend this result to the semigroup $\mathfrak U[F]$ consisting of all conformal…
We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a suffic…
We show that there exists a polynomial automorphism $f$ of $\mathbb{C}^{3}$ of degree 2 such that for every automorphism $g$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each $d \ge 2$, there exists an open set of polynomial automorphisms of degree at most $d$ in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz.
Dynamics of semigroups of entire maps of $\mathbb{C}^k$
本文研究了$\mathbb{C}^k,\; k \ge 2$的一类全纯自同态的Fatou集和朱利亚集的一些基本性质.我们特别感兴趣的研究这些集的半群生成的各种类的全纯自同态的$\mathbb{C}^k,\; k \ge 2.$我们证明了，如果朱利亚集的半群$G$生成的最大通秩$k$在$\mathbb{C}^k$的自同态包含一个孤立点，那么$G$必须包含一个元素，共轭的上三角自同构的$\mathbb{C}^k.$这推广了Fornaess-Sibony定理。其次，我们定义了半群的回归域，并在一定条件下给出了这类域的刻画。
The purpose of this article is to explore a few properties of polynomial shift-like automorphisms of $\mathbb{C}^k.$ We first prove that a $ν-$shift-like polynomial map (say $S_a$) degenerates essentially to a polynomial map in $ν-$dimensions as $a \to 0.$ Secondly, we show that a shift-like map obtained by perturbing a hyperbolic polynomial (i.e., $S_a$, where $|a|$ is sufficiently small) has finitely many Fatou components, consisting of basins of attraction of periodic points and the component at infinity.
We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.
For a Hénon map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a Hénon map turns out to be a Short $\mathbb{C}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short $\mathbb{C}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of …
We provide an experimental study of the existence of Herman rings in a parametrized family of rational maps preserving antipodal points, and a discussion of their properties on $\mathbb{RP}^2$. We study analytic maps of the sphere that project to dianalytic maps on the nonorientable surface, $\mathbb{RP}^2$. They have a known form and we focus on a subset of degree $3$ dianalytic maps and explore their dynamical properties. In particular, we focus on maps for which the Fatou set has a Herman ring. We appeal to dynamical properties of particular maps to justify our assertion that these Fatou co…
In this paper we study properties of endomorphisms of $\p^k$ using a symmetric product construction $(\p^1)^k/\mathfrak{S}_k \cong \p^k$. Symmetric products have been used to produce examples of endomorphisms of $\p^k$ with certain characteristics, $k\geq2$. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study $k$-deep post-critically finite maps and characterize families of Lattès maps.
We introduce a class of rational functions $A:\,\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ which can be considered as a natural extension of the class of Lattès maps and establish basic properties of functions from this class.
Let $IET(\mathbb{S}^{1})$ be the group of interval exchange transformation of $\mathbb{S}^{1}$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{ACI}$ the group generated by $IET(\mathbb{S}^{1})$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$. Given a suitable distance on $\mathcal{ACI}$, we classify all continuous homomorphisms $ρ:\mathbb{R} \to \mathcal{ACI}$. More precisely, $ρ$ is conjugated to a continuous homomorphism $\hatρ:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^…
We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space $\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitable partial order) and a "topological" condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronization of random orbits. As a consequence of the synchronization phenomenon, a functional central limit theorem is established.
We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_ζ$ that contains a Brjuno rotation $f_0(z)=z+α$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction co…
We prove that the dynamical system $(\mathbb{N}, 2^{\mathbb{N}}, T, μ)$, where $μ$ is a finite measure equivalent to the counting measure, is power-bounded in $L^1(μ)$ if and only if there exists one cycle of the map $T$ and for any $x \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that $T^k(x)$ is in some cycle of the map $T$. This result has immediate implications for the Collatz Conjecture, and we use it to motivate the study of number theoretic properties of the inverse image $T^{-1}(x)$ for $x \in \mathbb{N}$, where $T$ denotes the Collatz map here. We study similar properties for …
Let $q:M\to M$ be a volume-preserving diffeomorphism of a smooth manifold $M$. We study the possibility to present $q$ as the Poincaré map, corresponding to a volume-preserving vector field on $\mathbb{T}\times M$, $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.
We start the study of the family of birational maps $(f_{α,β})$ of $\mathbb{P}^2_\mathbb{C}$ in \cite{Deserti}. For generic $α$ and $β$ of modulus 1 the centraliser of $f_{α,β}$ is trivial, the topological entropy of $f_{α,β}$ is 0, there exist two areas of linearisation: in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On $\mathbb{P}^1_\mathbb{C}\times \mathbb{P}^1_\mathbb{C}$ any $f_{α,β}$ can be viewed as a cocyle; using recent results about $\mathrm{SL}(2;\mathbb{C})$-cocycles (\cite{Avila}) w…
We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of $\mathbb{C}^k$ such that $\mathrm{deg}\, f^n\sim n^\ell$. We also give counter-examples in dimension $k\geq 3$ to some classical properties satisfied by polynomial automorphisms of $\mathbb{C}^2$. We provide the existence of new degree growths in the context of birational maps of $\mathbb{P}^k_\mathbb{C}$: assume $k\geq 3$; forall $0\leq\ell\leq k$ there exist bi…
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $\mathbb{Z}^k$-actions which are generated by random compositions of the generators of $\mathbb{Z}^k$-actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $\mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $\mathbb{Z}^k$(or $\mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on fi…
For a circle map $f\colon\mathbb{S}\to\mathbb{S}$ with zero topological entropy, we show that a non-diagonal pair $\langle x,y\rangle\in \mathbb{S}\times \mathbb{S}$ is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.
In this manuscript we study rotationally $p$-harmonic maps between spheres. We prove that for $p\in\mathbb{N}$ given, there exist infinitely many $p$-harmonic self-maps of $\mathbb{S}^m$ for each $m\in\mathbb{N}$ with $p<m< 2+p+2\sqrt{p}$. In the case of the identity map of $\mathbb{S}^m$ we explicitly determine the spectrum of the corresponding Jacobi operator and show that for $p>m$, the identity map of $\mathbb{S}^m$ is stable when interpreted as a $p$-harmonic self-map of $\mathbb{S}^m$.
We study an action of ${\rm Aut}(F_n)$ on $\mathbb{R}^{2^n-1}$ by trace maps, defined using the traces of $n$-tuples of matrices in $\mathrm{SL}(2,\mathbb{C})$ having real traces. We determine the finite orbits for this action. These orbits essentially come from (i) the finite subgroups of $\mathrm{SL}(2,\mathbb C)$, and (ii) a dense set of (rational) points in an embedded quotient of an $n$-torus.
For any $d \geq 1$, random $\mathbb{Z}^d$ shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter $α\in [0,1]$, an alphabet $\mathcal{A}$, and a scale $n \in \mathbb{N}$, one obtains a distribution of random $\mathbb{Z}^d$ SFTs by randomly and independently forbidding each pattern of shape $\{1,\dots,n\}^d$ with probability $1-α$ from the full shift on $\mathcal{A}$. We prove two main results concerning random $\mathbb{Z}^d$ SFTs. First, we establish sufficient conditions on $α$, $\mathcal{A}$, and a $\mathbb{Z}^d$ subshift $Y$ so that a random $\mathbb{Z}^d$…
We prove the existence of invariant tori to the area-preserving maps defined on $ \mathbb{R}^2\times\mathbb{T} $ \begin{equation*} \bar{x}=F(x,θ), \qquad \barθ=θ+α\, \,(α\in \mathbb{R}\setminus\mathbb{Q}), \end{equation*} where $ F $ is closed to a linear rotation, and the perturbation is ultra-differentiable in $ θ\in \mathbb{T},$ which is very closed to $C^{\infty}$ regularity. Moreover, we assume that the frequency $α$ is any irrational number without other arithmetic conditions and the smallness of the perturbation does not depend on $α$. Thus, both the difficulties from the ultra-differen…
Let $f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $k\ge1$ and $\ell\ge0$ such that the critical locus $\operatorname{Crit}_f$ satisfies $f^{k+\ell}(\operatorname{Crit}_f)\subseteq{f^\ell(\operatorname{Crit}_f)}$. The smallest such $\ell$ is called the tail-length. We prove that for $d\ge3$ and $n\ge2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $\ell=0$, are not…
Jonsson and Reschke showed that birational selfmaps on projective surface defined over a number field satisfy the energy condition of Bedford and Diller so their ergodic properties are very well understood. Under suitable hypotheses on the indeterminacy loci, we extend that result to birational maps $\mathbb{P}^k\dashrightarrow\mathbb{P}^k$, $k\geq2$, defined over a number field, showing that they satisfy a similar energy condition introduced by De Thélin and the second author. As a consequence, we can construct for such maps their Green measure and deduce several important ergodic consequence…
In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. Given a sequence of degenerating rational maps, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the ultralimit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson's field. We show the two approaches are equivalent. The limiting dynamics on the $\mathbb{R}$-tree are analogues to i…
Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{ϕ\in\text{PGL}_3(K): ϕ^{-1}\circ f\circϕ=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.
Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. In this article, we introduce two new equivariants associated to a rational map $f\in K(z)$. These objects naturally live on a real hyperbolic space, and carry information about the action of $f$ on $\mathbb{P}^1(K)$. When $K=\mathbb{C}$ we relate the asymptotic behavior of these equivariants to the conformal barycenter of the measure of maximal entropy. We also give a complete description of these objects for rational maps of degree $d=1$. The constructions in this article are based on work of Rumely in the context of rational maps over non-…
We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\in\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\mathrm{Per}_n(0)$ is the $1$-dimensional moduli space of quadratic rational maps on $\mathbb{P}^1$ with an $n$-periodic critical point. We show that if $G_n$ is irreducible over $\mathbb{Q}$, then $\mathrm{Per}_n(0)$ is irreducible over $\mathbb{C}$. To do this, we exhibit a $\mathbb{Q}$-rational smooth point on a projective completion of…
