arXiv subjects

Hiroki Sumi

Publications and source records attributed to Hiroki Sumi.

At least 37 records · Page 2Linked to original sources

arXiv2016

The space of $2$-generator postcritically bounded polynomial semigroups and random complex dynamics

We investigate the dynamics of $2$-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space ${\cal B}$ of such semigroups. We show that for a parameter $h$ in the intersection of ${\cal B}$, the hyperbolicity locus ${\cal H}$ and the closure of the disconnectedness locus (the space of parameters for which the Julia set is disconnected), the corresponding semigroup satisfies either the open set condition (and the Bowen's formula) or that the Julia sets of the two generators coincide. Also, we show that for such a parameter $h$, if the Julia sets of the two generators do not coincide, then there exists a neighborhood $U$ of $h$ in the full parameter space ${\cal P}^{2}$ such that for each parameter in $U$, the Hausdorff dimension of the Julia set of the corresponding semigroup is strictly less than 2. Moreover, we show that the intersection of the connectedness locus and ${\cal B}\cap {\cal H}$ has dense interior. By using the results on the semigroups corresponding to these parameters, we investigate the associated functions which give the probability of tending to $\infty$ (complex analogues of the devil's staircase or Lebesgue's singular functions) and complex analogues of the Takagi function.

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arXiv2015

Multifractal formalism for expanding rational semigroups and random complex dynamical systems

We consider the multifractal formalism for the dynamics of semigroups of rational maps on the Riemann sphere and random complex dynamical systems. We elaborate a multifractal analysis of level sets given by quotients of Birkhoff sums with respect to the skew product associated with a semigroup of rational maps. Applying these results, we perform a multifractal analysis of the Hölder regularity of limit state functions of random complex dynamical systems.

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arXiv2015

Random complex dynamics and devil's coliseums

We investigate the random dynamics of polynomial maps on the Riemann sphere and the dynamics of semigroups of polynomial maps on the Riemann sphere. 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 $T_{\infty}$ of probability of tending to $\infty$ is Hölder continuous on the Riemann sphere and varies only on the Julia set of $G$. Moreover, the function $T_{\infty}$ has a kind of monotonicity. It turns out that $T_{\infty}$ is a complex analogue of the devil's staircase, and we call $T_{\infty}$ a "devil's coliseum." We investigate the details of $T_{\infty}$ when $G$ is generated by two polynomials. In this case, $T_{\infty}$ varies precisely on the Julia set of $G$, which is a thin fractal set. Moreover, under this condition, we investigate the pointwise Hölder exponents of $T_{\infty}$ by using some geometric observations, ergodic theory, potential theory and function theory. In particular, we show that for almost every point $z$ in the Julia set of $G$ with respect to an invariant measure, $T_{\infty}$ is not differentiable at $z.$ We find many new phenomena of random complex dynamics which cannot hold in the usual iteration dynamics of a single polynomial, and we systematically investigate them.

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arXiv2013

Random Backward Iteration Algorithm for Julia sets of Rational Semigroups

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.

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arXiv2013

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 $J_γ$ to satisfy that $J_γ$ is a Jordan curve but not a quasicircle, the unbounded component of the complement of $J_γ$ is a John domain and the bounded component of the complement of $J_γ$ 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.

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arXiv2013

Cooperation principle, stability and bifurcation in random complex dynamics

We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, even though each map of the system has a chaotic part. Moreover, we show that (1) the set of stable systems is open and dense in the space of random dynamical systems of polynomials, and (2) for a stable system, there exist at most finitely many minimal sets and each minimal set is attracting. These (1) and (2) solve a random analogy of the most famous open conjecture in complex dynamics "hyperbolic rational maps are dense in the space of rational maps." Furthermore, we show that for a stable system, the orbit of a Hölder continuous function under the transition operator tends exponentially fast to the finite-dimensional space $U$ of finite linear combinations of unitary eigenvectors of the transition operator. From this, we obtain that the spectrum of the transition operator acting on the space of Hölder continuous functions has a gap between the set of unitary eigenvalues and the rest. Combining this with the perturbation theory for linear operators, we obtain that for a stable system constructed by a finite family of rational maps, the projection to the space $U$ depends real-analytically on the probability parameters. By taking a partial derivative of the function of probability of tending to a minimal set with respect to a probability parameter, we introduce a complex analogue of the Takagi function, which is a new concept. Many new phenomena which can hold in random complex dynamics but cannot hold in usual iteration dynamics of a single rational map are found and systematically investigated.

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arXiv2013

Transversality Family of Expanding Rational Semigroups

This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2. We also show that a family of small perturbations of Sierpiński gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each complex number $a$ with $|a|\neq 0,1$, the family of small perturbations of the semigroup generated by ${z^{2}, az^2} $ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.

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arXiv2012

Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We estimate the Bowen parameters (zeros of the pressure functions) and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $F$ and the Lyapunov exponent of $F$ with respect to the maximal entropy measure for $F$. Moreover, we show that the equality holds if and only if the generators $f_{j}$ are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a linear fractional transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than two.

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arXiv2011

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) dynamics, 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 finite critical value of any map $g \in G$. 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 (\cite{Su01,Su9}). 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 $g \in G$ 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.

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arXiv2011

Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.

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arXiv2011

Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any $n\in \Bbb{N} \cup \{\aleph_{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated.

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arXiv2011

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) 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 $γ\in Γ^{\Bbb{N}}$, the Julia set $J_γ$ of $γ$ is a Jordan curve but not a quasicircle, the unbounded component of the Fatou set $F_γ$ of $γ$ is a John domain, and the bounded component of $F_γ$ 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.

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arXiv2011

Random complex dynamics and semigroups of holomorphic maps

We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, "singular functions on the complex plane" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity with respect to the random dynamics of polynomials. Under certain conditions these functions $T$ are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions $T$ are continuous on the Riemann sphere and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the non-differentiability and regularity of these functions. We find many phenomena which can hold in the random complex dynamics and the dynamics of semigroups of rational maps, but cannot hold in the usual iteration dynamics of a single holomorphic map. We carry out a systematic study of these phenomena and their mechanisms.

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arXiv2010

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 Bowen's parameters and Hausdorff dimension function.

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arXiv2009

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings${\cal U} $ of the invariant set, where each ${\cal U}$ consists of (backward)images of the invariant set under elements of finite word length. Inparticular, we give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of finitely generated semigroups of polynomials on the complex plane. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.

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arXiv2009

The equilibrium states for semigroups of rational maps

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.

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arXiv2008

Analytic families of holomorphic iterated function systems

This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the "$λ$-topology". This classification theorem allows us to generalize some geometric results from the paper ("Lambda-Topology vs. Pointwise Topology", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clearer understanding of the global structure of the space of conformal IFSs.

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arXiv2007

Dynamics of postcritically bounded polynomial semigroups

We investigate the dynamics of semigroups generated by a family of 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. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. Moreover, we show that for any $n\in \Bbb{N} \cup \{\aleph_{0}\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Furthermore, we show that under a certain condition, 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.

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