The symmetric group S_n acts as a reflection group on CP^{n-2} (for $n\geq 3$) . Associated with each of the $\binom{n}{2}$ transpositions in S_n is an involution on CP^{n-2} that pointwise fixes a hyperplane--the mirrors of the action. For each such action, there is a unique S_n-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations of symmetry and critical-finiteness produce global dynamical results: each map's fatou set consists of a special finite set of superattracting points whose basins are dense.
We discuss the dynamical, topological, and algebraic classification of rational maps $f$ of the Riemann sphere to itself each of whose critical points $c$ is also a fixed-point of $f$, i.e. $f(c)=c$.
We study the dynamics of post-critically finite endomorphisms of P^k(C). We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a mild regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a weak transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins, thus partially extending to any dimension a result of Fornaess-Sibony and Ron…
It is a classical result in complex dynamics of one variable that the Fatou set for a critically finite map on $\mathbf{P}^1$ consists of only basins of attraction for superattracting periodic points. In this paper we deal with critically finite maps on $\mathbf{P}^k$. We show that the Fatou set for a critically finite map on $\mathbf{P}^2$ consists of only basins of attraction for superattracting periodic points. We also show that the Fatou set for a $k-$critically finite map on $\mathbf{P}^k$ is empty.
We study eigenvalues along periodic cycles of post-critically finite endomorphisms of $\mathbb{CP}^n$ in higher dimension. It is a classical result when $n = 1$ that those values are either $0$ or of modulus strictly bigger than $1$. It has been conjectured in [Van Tu Le. Periodic points of post-critically algebraic holomorphic endomorphisms, Ergodic Theory and Dynamical Systems, pages 1-33, 2020] that the same result holds for every $n \geq 2$. In this article, we verify the conjecture for the class of weakly post-critically finite all the way down maps which was introduced in [Matthieu Astor…
In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence…
This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f^n is the subdivision map of a finite subdivision rule.
In this paper, 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 establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex numbers {\bf C} to itself which have degree two or more in each copy. As a consequence of these results we prove a transitivity relation between hyperbolic components in parameter space which was conjectured by Milnor.
We prove: If $f(z)$ is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of $f$ is a Jordan curve. If $f(z)$ is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps $f$ with the property that on the closure of a Fatou component $Ω$ satisfying $f(Ω)=Ω$, $f|_{\bdry Ω}$ is not topologically conjugate to the dyn…
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…
The theory of Hubbard trees provides an effective classification of non-linear post-critically finite polynomial maps from \C to itself. This note will extend this classification to the case of maps from a finite union of copies of \C to itself. Maps which are post-critically finite and nowhere linear will be characterized by a ``forest'', which is made up out of one tree in each copy of \C.
A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the…
We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finit…
We consider random multimodal $C^3$ maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $[0,1]$, onto the interval $[0,1]$ with the base space $Ω$ and a base invertible ergodic map $θ:Ω\toΩ$ preserving a probability measure $m$ on $Ω$. We denote the corresponding skew product map by $T$ and call it a critically finite random map of an interval. We prove that there exists a subset $AA(T)$ of $[0,1]$ with the following properties: (1) For each $t\in AA(T)$ a $t$-conformal random measure $ν_t$ exists. We denote by $λ_{t,ν_t,ω}$ the corresponding generalized …
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.
A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one exceptional badly behaved case, and provide a new description of the appropriate moduli spaces. There is also an application to topological entropy.
We study the dynamics of a family of continued fraction maps parametrized by the unit interval. This family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these dynamical maps and make a preliminary study of them. We show that their analytic invariant measures obeys a common functional equation generalizing Lewis' functional equation and we find invariant measures for some members of the family. We also discuss a certain involution of this family which sends the Gauss map to the Fibonacci map.
This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim in [4] as one of its subfamilies.
We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \ge 1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.
Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to the Julia set of $Q$. We use this fact to compute analytic invariants of the nest of $f_c$, to give a complete characterization of complex quadratic Fibonacci maps and to obtain a new auto-similarity result on the Mandelbrot set.
We give a combinatorial classification of postsingularly finite exponential maps in terms of external addresses starting with the entry 0. This is an extension of the classification results for critically preperiodic polynomials \cite{BFH} to exponential maps. Our proof relies on the topological characterization of postsingularly finite exponential maps given recently in \cite{HSS}. Our results illustrate once again the fruitful interplay between combinatorics, topology and complex structure which has often been successful in complex dynamics.
We investigate the break up of the last invariant curve for families of standard maps. Our main result is another evidence of how hard this problem is. We give an example of an analytic mapping with a "pathological" behavior, related to the break up of the last rotational invariant curve.
Continuing work begun in a previous paper, we study the real dynamics of a family of plane birational maps. This time we consider a parameter range where the real and complex dynamics are different. We show in particular that for one parameter value, the real dynamics are transient even though the map is not integrable.
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of dimension 1, a deep result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattes maps. This note proves a corresponding result in arbitrary dimension, for a certain subclass of morphism. Specifically, we restrict attention to morphisms f of degree at least two, with…
We consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching
The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals 1 for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals 1. However, the behaviour of the determinism in the infinite horizon is counter-in…
Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as maps with "contracting biset". We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Ree…
It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least 5. This paper introduce a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia compo…
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get the same results for more general families of unimodal maps.
From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point $b$ corresponding to a Nash equilibrium of such map $f$ is usually repelling, it is globally Cesaro attracting on the diagonal, that is, \[ \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f^k(x)=b \] for every $x$ in the minimal invariant interval. Thi…
In this note we consider a collection $\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\{μf\}$ where $μ\in [0,1].$ Denoting the kneading sequence of $μf$ by $K(μf)$, we will prove that for each member of $\cal{C}$, the map $μ\mapsto K(μf)$ is monotone. It then follows that for each member of $\cal{C}$ the map $μ\mapsto h(μf)$ is monotone, where $h(uf)$ is the topological entropy of $μf.$ For interest, $μf(x)=4μx(1-x)$ and $μf(x)=μ\sin(πx)$ are shown to belong to $\cal{C}.$ This extends the work of Masato Tsujii [1].
For a two parameter family of two-dimensional piecewise linear maps and for every natural number $ n $ we prove not only the existence of intervals of parameters for which the respective maps are $ n $ times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $ 2^n $ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.
Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $μ_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $Γ(t)=\int ϕdμ_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $Γ(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continu…
We define a family of discontinuous maps on the circle, called Bowen-Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor-Thurston theory for one dimensional maps.
We study the exponentially small splitting of separatrices in an analytic one-parameter family of area-preserving maps that generically unfolds a 1:3 resonance. Near the resonance the normal form theory predicts existence of a small triangle formed by separatrices of a period three hyperbolic point. We prove that in a generic family the separatrices split, provided that the Stokes constant of the map does not vanish. This constant describes the distance between the analytical continuations of invariant manifolds associated with the degenerate saddle of the map at the exact resonance. We provid…
Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计学院）；Instituto de Matemática Pura e Aplicada（纯数学与应用数学研究所）；Shenzhen International Center for Mathematics, SUSTech（南方科技大学深圳国际数学中心）
The dynamics of a 1-parameter family of cluster maps $φ_r$ associated to mutation-periodic quivers in dimension 4, is studied in detail. The use of presymplectic reduction leads to a globally periodic symplectic map, and this enables us to reduce the problem to the study of maps belonging to a group of symplectic birational maps of the plane which is isomorphic to $SL(2,\mathbb{Z})\ltimes\mathbb{R}^2$. We conclude that there are three different types of dynamical behaviour for $φ_r$ characterized by the integer parameter values $r=1$, $r=2$ and $r>2$. For each type, the periodic points, the st…
Intermittent maps of Pomeau-Manneville type are well-studied in one-dimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theore…
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical co…
We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the `stretching and folding' action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well known systems, Arnold's Cat Map and a map due to Cerbelli and Giona, both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such fin…
Let $\{f_t\}_{t\in(1,2]}$ be the family of core tent maps of slopes $t$. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $Φ_t\colon D^2\to D^2$, having transitive global attractors $Λ_t$ on which $Φ_t$ is topologically conjugate to the natural extension of $f_t$. The unique family of absolutely continuous invariant measures for $f_t$ induces a family of ergodic $Φ_t$-invariant measures $ν_t$, supported on the attractors $Λ_t$. We show that this family $ν_t$ varies weakly continuously, and that the measures $ν_t$ are physical with respect to a weakly continuou…
In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem.
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of that entropy with respect to the parameter. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of t…
'Generalized del Junco-Rudolph's map', a sub-family of generalized Chacon's map (\cite{FER1}), is introduced. A skew product related to the structure of the Generalized del Junco-Rudolph's map is introduced. A Relative Prime Relation, $h_{k+1}\equiv 1 \mod q$ is verified, based on the proposition of iterations of this skew product. We say a measure-preserving transformation $T$ is totally rank one if $T^{q}$ is rank one for every $q\in \mathbb{N}$. In this paper, we show that of every Generalized del Junco-Rudolph's map is totally rank one.
We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.
A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.
We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Hénon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.
We compute the degree complexity of the family of birational maps considered in \cite{bedford-kim-tuyen-abarenkova-maillard} for all exceptional cases. Some interesting properties of the family are also given.
We consider the logistic family and apply the $q$-deformation $ϕ_q(x)=\frac{1-q^x}{1-q}$. We study the stability regions of the fixed points of the $q$-deformed logistic map and the regions where the dynamic is complex through topological entropy and Lyapunov exponents. Our results show that the dynamic of this deformed family is richer than that of the $q$-deformed family studied in [8].
In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which the corresponding map is post-critically finite with a certain dynamical portrait. Then we prove that the polynomial is irreducible in certain cases.
We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function cla…
