Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $μ_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapstoμ_f$ is known to be continuous on $Rat_d$, and it is shown here to extend continuously to the boundary of $Rat_d$ in $\bar{Rat}_d \simeq {\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\mapsto f^n$ for every $n\geq 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral.
We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely generated rational semigroups (of a certain type). We also provide some consequences of this result.
The iteration of rational maps is well-understood in dimension 1 but less so in higher dimensions. We study some maps on spaces of matrices which present a weak complexity with respect to the ring structure. First we give some properties of certain rational maps; the simplest example is the rational map which sends the matrix $\mathrm{M}$ onto $\mathrm{M}^2$ for which we exhibit some dynamical properties. Finally we deal with some small perturbations of this map.
We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role in the study of general Siegel rational maps. As one of the applications, we prove that for any quadratic rational map with a bounded type Siegel disk, the boundary of the Siegel disk is a quasi-circle which passes through one or both of the critical points.
This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the type…
The aim of these lectures is the study of bifurcations within holomorphic families of polynomials or rational maps by mean of ergodic and pluripotential theoretic tools.
For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the set of fixed point free rational self-maps of $\pp^2$ is closed (modulo "degenerate" maps) in some natural topology on the space of rational self-maps of $\pp^2$.
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.
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.
Let $f$ be a degree $d$ bicritical rational map with critical point set $\mathcal{C}_f$ and critical value set $\mathcal{V}_f$. Using the group $\textrm{Deck}(f^k)$ of deck transformations of $f^k$, we show that if $g$ is a bicritical rational map which shares an iterate with $f$ then $\mathcal{C}_f = \mathcal{C}_g$ and $\mathcal{V}_f = \mathcal{V}_g$. Using this, we show that if two bicritical rational maps of even degree $d$ share an iterate then they share a second iterate, and both maps belong to the symmetry locus of degree $d$ bicritical rational maps.
We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks.
Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${ń}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${ń}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.
Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic rational map having exactly two critical grand orbits but also having a Herman ring. In particular, $f$ has no Herman rings if it has at most one infinite critical orbit in the Julia set. These criterions derive some known results about the rational maps without Herman rings.
We show that there is a spectral gap for the transfer operator associated to a rational map f on the Riemann sphere. Using this and the method of pertubed operators we establish the (Local) Central Limit Theorem for the measure of maximal entropy of f, with an estimate on the speed of convergence.
In this paper, we study Markovian random iterations of maps on standard measurable spaces. We establish a one-to-one correspondence between stationary measures and a certain class of invariant measures of a Markovian random iteration, extending a similar classical result of independent and identically distributed random iterations. As an application, we prove a local synchronization property for Markovian random iterations of homeomorphisms of the circle $S^{1}$.
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several new periodic quivers.
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.
In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled ``On Bicritical Rational Maps'' in September 1997; revised and retitled April 1999)
We show that the set of Misiurewicz maps has Lebesgue measure zero in the parameter space of rational maps for any fixed degree greater than or equal to 2.
We use intuitive results from algebraic topology and intersection theory to clarify the pullback action on cohomology by compositions of rational maps. We use these techniques to prove a simple sufficient criterion for functoriality of a composition of two rational maps on all degrees of cohomology and we then reprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We conclude with a cautionary example.
Let $\{N_t\}$ be a holomorphic family of degree $d\ge 3$ Newton maps. By studying the related Berkovich dynamics, we obtain an estimate of the weak limit of the maximal measures of $N_t$. Moreover, we give a complete description of the rescaling limits for $\{N_t\}$.
We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.
We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function ψ, there exists a unique equilibrium state for ψand a unique $\exp(¶(ψ)-ψ)$-conformal measure, where P(ψ) denotes the topological pressure of ψ.
If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.
In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.
In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.
Let ${\cal H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that ${\cal H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
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…
In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.
We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree greater than or equal to 2 (generalising the earlier version math.DS/0701382).
We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.
In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.
A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.
We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space.It is shown that any given sequence of maps of degree $d \ge 2$ has at most $2d-2$ dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained employing tools from non-Archimedean dynamics.
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps …
We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.
We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which to conjugate any degree two rational map corresponding to an integral point in the moduli space into a map with unit resultant. The local fields result is used to solve the same problem for fields over a principal ideal domain. Some additional results are given for degree 2 rational maps over the rationals.
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.
We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain $σ$-finite conformal measures. This allows to extend previous results to exceptional rational maps.
The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps admitting antiholomorphic automorphisms. The locus of the real points ${\rm M}_{d}({\mathbb R})$ decomposes as a disjoint union of the loci ${\rm M}_{d}^{\mathbb R}$, consisting of the real rational maps, and ${\mathcal P}_{d}$, consisting of the pseudo-real ones. We obtain that, both ${\rm M}_{d}^{\mathbb R}$ and ${\rm M}_{d}({\mathbb R})$, are connected and…
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. $f(c)=c$.
The aim of this paper is to show $J$-stability of immediately expanding rational maps over an algebraically closed, complete, and non-Archimedean field, which is an analogue of R. Mane, P. Sad, and D. Sullivan's theorem of $J$-stability in complex dynamical systems.
This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results about nonrational critically finite degree two branched coverings, and finally identify the boundary of the rational maps in the combinatorial model, thus completing the proofs of results announced in Part 1.
We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou components are Jordan disks with disjoint closures (Sierpiński maps), as well as those where any two Fatou components can be connected through a countable chain of Fatou components with common boundary points (crochet or Newton-like maps). We provide several alternative characterizations for our decomposition, as well as an algorithm for its effective computation. …
In this paper we investigate the perturbation properties of rational Misiurewicz maps, when the Julia set is the whole sphere (the other case is treated in [1]). In particular, we show that if f is a Misiurewicz map and not a flexible Lattes map, then we can find a hyperbolic map arbitrarily close to f.
