Periodic Critical Points Research Articles

Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps

We study the balanced resistance forms on the Julia sets of Misiurewicz-Sierpinski maps, which are self-similar resistance forms with equal weights. In particular, we use a theorem of Sabot to prove the existence and uniqueness of balanced forms on these Julia sets. We also provide an explorative study on the resistance forms on the Julia sets of rational maps with periodic critical points.

Algorithmic aspects of branched coverings II/V: sphere bisets and decidability of Thurston equivalence

We consider Thurston maps: branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are sphere trees of bisets. To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer—while centralizers of homeomorphisms are always finitely generated.

Critically-Finite Dynamics on the Icosahedron

A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes the presentation of which concludes the paper.

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ together with a real-valued Holder continuous potential $${\phi}$$ . Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.

On periodic critical points and local minimizers of the Ohta–Kawasaki functional

In this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem arising in the modeling of diblock copolymers. In the main result, by means of a purely variational procedure, we show that it is possible to construct (locally minimizing) periodic critical points whose shape resemble that of any given strictly stable constant mean curvature (periodic) hypersurface. Along the way, we establish several auxiliary results of independent interest.

Weak expansion properties and large deviation principles for expanding Thurston maps

In this paper, we prove that an expanding Thurston map f:S2→S2 is asymptotically h-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is h-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on S2, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map without periodic critical points and a potential that is Hölder continuous with respect to a visual metric on S2.

The set of dynamically special points

We call a point “dynamically special” if it has a dynamical property, which no other point has. We prove that, for continuous self maps of the real line, all dynamically special points are in the closure of the union of the full orbits of periodic points, critical points and limits at infinity. We completely describe the set of dynamically special points of real polynomial functions. The following characterization for the set of special points is also obtained: A subset of \({\mathbb{R}}\) is the set of dynamically special points for some continuous self map of \({\mathbb{R}}\) if and only if it is closed.

A case of the dynamical Mordell–Lang conjecture

We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on $${(\mathbb P^1)^g}$$ . If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of $${(\mathbb P^1)^g}$$ has only finite intersection with any curve contained in $${(\mathbb P^1)^g}$$ . We also show that our result holds for indecomposable polynomials φ with coefficients in $${\mathbb C}$$ . Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over $${\mathbb C}$$ uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ, φ) on $${\mathbb A^2}$$ .

Measure preserving transformations similar to Markov shifts

Similarity, that is, the existence of joint common extensions, defines an interesting equivalence relation for infinite measure preserving transformations T. We provide a sufficient condition, given in terms of return processes to reference sets of finite measure, for T to be similar to a Markov shift. This is then shown to apply to various piecewise smooth dynamical systems, including weakly hyperbolic transformations with indifferent periodic points or flat critical points, and discrete random walks driven by (weakly) hyperbolic maps.

Constructing subdivision rules from rational maps

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n n the iterate f ∘ n f^{\circ n} is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates.

A classification of inverse limit spaces of tent maps with periodic critical points

We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps fa, fb with periodic critical points, we show that the inverse limit spaces (Ia;fa) and (Ib;gb) are not homeomorphic when a 6 b. To obtain our result, we dene topological substructures of a composant, called \wrapping points and \gaps, and identify properties of these substructures preserved under a homeomorphism.

Synergy, sigmoids and the seventh-year trifurcation

Summary Whereas linear and exponential growth have become relatively familiar metaphors, second-order growth processes, as exemplified by autocatalytic chemical reactions, have not yet entered the vernacular. Derek J. de Solla Price has discussed sigmoid growth, but described it as a three-phase process. It is shown here that these three phases are more apparent than real, with initial slow growth, intermediate rapid growth and ultimate saturation all characteristic of the same second-order process. A model, arrived at by the superposition of several sigmoid curves, is proposed for the periodic critical decision points which occur during the course of a career or the life time of an organization. These decision points correspond to a trifurcation, leading to renewed sigmoidal growth, saturation, or a rapid phase-out.