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Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 30 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Random complex dynamics and semigroups of holomorphic maps
, 2008
"... We investigate the random dynamics of rational maps on the Riemann sphere Ĉ and the dynamics of semigroups of rational maps on Ĉ. We see that the both fields are related to each other very deeply. We investigate spectral properties of transition operators and the dynamics of associated semigroups of ..."
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Cited by 15 (11 self)
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We investigate the random dynamics of rational maps on the Riemann sphere Ĉ and the dynamics of semigroups of rational maps on Ĉ. We see that the both fields are related to each other very deeply. We investigate spectral properties of transition operators and the dynamics of associated semigroups of rational maps. We define several kinds of Julia sets of the associated Markov processes and we study the properties and the dimension of them. Moreover, we investigate “singular functions on the complex plane”. 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 Ĉ and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the nondifferentiability 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.
Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups
, 2009
"... 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 con ..."
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Cited by 12 (12 self)
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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 ∈ 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 ([29, 33]). 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 ∈ 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.
Measures and dimensions of Julia sets of semihyperbolic rational semigroups
, 2008
"... We consider the dynamics of semihyperbolic 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 ..."
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We consider the dynamics of semihyperbolic 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 hdimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with RadonNikodym 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 skewproduct map there exists a unique hconformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skewproduct map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.