Hiroki Sumi

Skew product maps related to finitely generated rational semigroups

We consider skew-product maps related to dynamics of semigroups generated by rational maps on the Riemann sphere. The entropy of these maps will be given and we will see there exists the unique maximal entropy measure. We will also show the uniqueness of the self-similar measure. We will estimate the Hausdorff dimension of the Julia sets of semigroups.

Semi-hyperbolic fibered rational maps and rational semigroups

We consider fiber-preserving complex dynamics on fiber bundles whose fibers are the Riemann spheres and whose base spaces are com- pact metric spaces. We define the semi-hyperbolicity of dynamics on fiber bundles. We will show that if adynamics on fiber bundle is semi- hyperbolic, then we have that the fiberwise Julia sets are k-porous and that the dynamics has akind of weak rigidity. We also show that the Julia set of rational semigroup(semigroup generated by rational maps

Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles ⁄

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere ˆ C) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence γ ∈ Γ N , the Julia set Jγ is a Jordan curve but not a quasicircle, the unbounded component of ˆ C \ Jγ is a John domain, and the bounded component of C \ Jγ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynam- ics, where the semigroup G of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any flnite critical value of any map g 2 G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G: Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps g 2 G are distributed within the Julia set of the entire semigroup G. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition.

Random dynamics of polynomials and devil’s-staircase-like functions in the complex plane

We consider the dynamics of polynomial semigroups with bounded postcritical set and random dynamics of complex polynomials in the complex plane. A polynomial semigroup G is a semigroup generated by polynomials in one variable with the semigroup operation being functional composition. We show that if the postcritical set of G, that is the closure of the G-orbit of the union of any critical values of any generators of G, is bounded in the complex plane, then the space of components of the Julia set of G (Julia set is the set of points in the Riemann sphere C¯ in which G is not normal) has a total order “⩽”, where for two compact connected sets K1, K2 in C¯, K1 ⩽ K2 indicates that K1 = K2, or K1 is included in a bounded component of C¯⧹K2. Using the above result and combining it with the theory of random dynamics of complex polynomials, we consider the following: Let τ be a Borel probability measure in the space {g∈C[z]|deg(g)⩾2} with topology induced by the uniform convergence on the Riemann sphere C¯. We consider the i.i.d. random dynamics in C¯ such that at every step we choose a polynomial according to the distribution τ. Let T∞(z) be the probability of tending to ∞∈C¯ starting from the initial value z∈C¯ and let Gτ be the polynomial semigroup generated by the support of τ. Suppose that the support of τ is compact, the postcritical set of Gτ is bounded in the complex plane and the Julia set of Gτ is disconnected. Then, we show that (1) in each component U of the complement of the Julia set of Gτ, T∞∣U equals a constant CU, (2) T∞:C¯→[0,1] is a continuous function on the whole C¯, and (3) if J1, J2 are two components of the Julia set of Gτ with J1 ⩽ J2, then maxz∈J1T∞(z)⩽minz∈J2T∞(z). Hence T∞ is similar to the devil’s-staircase function.

Real analyticity of Hausdorff dimension for expanding rational semigroups

We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic families of semigroups satisfying all the above conditions and we analyze in detail the corresponding Bowens parameters and Hausdorff dimension function.

Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set Jγ to satisfy that Jγ is a Jordan curve but not a quasicircle, the unbounded component of ˆ C \ Jγ is a John domain and the bounded component of C \ Jγ is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.

Random complex dynamics and devil's coliseums ⁄

We investigate the random dynamics of polynomial maps on the Riemann sphere ˆ C and the dynamics of semigroups of polynomial maps on ˆ C. In particular, the dynamics of a semigroup G of polynomials whose planar postcritical set is bounded and the associated random dynamics are studied. In general, the Julia set of such a G may be disconnected. We show that if G is such a semigroup, then regarding the associated random dynamics, the chaos of the averaged system disappears in the C 0 sense, and the function T∞ of probability of tending to ∞ ∈ ˆ C is continuous on ˆ C and varies only on the Julia set of G. Moreover, the function T∞ has a kind of monotonicity. It turns out that T∞ is a complex analogue of the devil’s staircase, and we call T∞ a “devil’s coliseum.” We investigate the details of T∞ when G is generated by two polynomials. In this case, T∞ varies precisely on the Julia set of G, which is a thin fractal set.

Analytic families of holomorphic iterated function systems

This paper deals with analytic families of holomorphic iterated function systems (IFSs). Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic IFSs which depend continuously on a parameter when the space of holomorphic IFSs is endowed with the λ-topology. This classification theorem allows us to generalize some geometric results from [16] and gives us a better and clearer understanding of the global structure of the space of conformal IFSs.

Lambda-topology versus pointwise topology

This paper deals with families of conformal iterated function systems (CIFSs). The space CIFS( X , I ) of all CIFSs, with common seed space X and alphabet I , is successively endowed with the topology of pointwise convergence and the so-called λ -topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense G δ -set, it is also discontinuous on a dense subset of CIFS( X , I ). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS( X , I ), which goes against intuition and conception of a natural topology on CIFS( X , I ). We then prove how good the λ -topology is: Roy and Urbanski [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. & Dynam. Sys. 25 (6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS( X , I ). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ -topology makes the space CIFS( X , I ) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS( X , I ) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS( X , I ).