DCDS
Bowen parameter and Hausdorff dimension for expanding rational semigroups
Hiroki Sumi Mariusz Urbański
Discrete & Continuous Dynamical Systems - A 2012, 32(7): 2591-2606 doi: 10.3934/dcds.2012.32.2591
We estimate the Bowen parameters 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 $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.
keywords: Bowen parameter Hausdorff dimension random complex dynamics. Julia set rational semigroups Complex dynamical systems expanding semigroups
DCDS
Random backward iteration algorithm for Julia sets of rational semigroups
Rich Stankewitz Hiroki Sumi
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2165-2175 doi: 10.3934/dcds.2015.35.2165
We provide proof that a random backward iteration algorithm for approximating Julia sets of rational semigroups, previously proven to work in the context of iteration of a rational function of degree two or more, extends to rational semigroups (of a certain type). We also provide some consequences of this result.
keywords: random complex dynamics Rational semigroups Julia sets random iteration invariant measure. Markov process
DCDS
Backward iteration algorithms for Julia sets of Möbius semigroups
Rich Stankewitz Hiroki Sumi
Discrete & Continuous Dynamical Systems - A 2016, 36(11): 6475-6485 doi: 10.3934/dcds.2016079
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [8].
keywords: random complex dynamics Rational semigroups random iteration Julia sets Markov process Möbius maps invariant measure.
DCDS
Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups
Hiroki Sumi Mariusz Urbański
Discrete & Continuous Dynamical Systems - A 2011, 30(1): 313-363 doi: 10.3934/dcds.2011.30.313
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.
keywords: Complex dynamical systems rational semigroups Julia set Hausdorff dimension skew product. semi-hyperbolic semigroups conformal measure
DCDS
Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets
Hiroki Sumi
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1205-1244 doi: 10.3934/dcds.2011.29.1205
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 $n\in N \cup \{ \aleph _{0}\} ,$ 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.
keywords: Julia set surrounding order. polynomial semigroup iterated function systems rational semigroup fractal geometry random iteration random complex dynamical systems Complex dynamical systems

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