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68
Conformal Iterated Function Systems With Applications To The Geometry Of Continued Fractions
, 1998
"... . In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set o ..."
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Cited by 30 (9 self)
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. In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries. 1 Research supported by NSF Grant DMS9502952. AMS(MOS) subject classifications(1980). Primary 28A80; Secondary 58F08, 58F11, 28A78 Key words and phrases. Iterated function systems, continued fractions, Hausdorff dimension, Hausdorff and packing measures, arithmetic densities. Typeset by A M ST E X Mauldin and Urba'nski Page 1 x1. Introduction: Setting and Notation Let I be a nonempty subset of N , the se...
of Julia sets of expanding rational semigroups
 See also http://arxiv.org/abs/math.DS/0405522.) 30 R.Stankewitz, T.Sugawa, H.Sumi, Some counterexamples in dynamics of rational semigroups, Annales Academiae Scientiarum Fennicae Mathematica Vol.29
"... We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finit ..."
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Cited by 13 (10 self)
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We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen’s formula, and the existence and uniqueness of a conformal measure, for a finitely generated expanding semigroup satisfying the open set condition. 1
Conformal Measures for Rational Functions Revisited
, 1998
"... We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by ap ..."
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Cited by 11 (1 self)
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We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
, 2007
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Dimension And Measures For A Curvilinear Sierpinski Gasket Or Apollonian Packing
, 1998
"... . In this paper we apply some results about general conformal iterated function systems to A, the residual set of a standard Apollonian packing or a curvilinear Sierpinski gasket. Within this context, it is straight forward to show that h; the Hausdorff dimension of A is greater than 1 and the packi ..."
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Cited by 8 (2 self)
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. In this paper we apply some results about general conformal iterated function systems to A, the residual set of a standard Apollonian packing or a curvilinear Sierpinski gasket. Within this context, it is straight forward to show that h; the Hausdorff dimension of A is greater than 1 and the packing dimension and the upper and lower box counting dimensions are all the same as the Hausdorff dimension. Among other things, we verify Sullivan's result that 0 ! H h (A) ! 1 and P h (A) = 1: 1 Research supported by NSF Grant DMS9502952. AMS(MOS) subject classifications(1980). Primary 28A80; Secondary 58F08, 58F11, 28A78 Key words and phrases. Apollonian packing, Sierpinski gasket, iterated function systems, Hausdorff dimension, Hausdorff and packing measures. Typeset by A M ST E X Mauldin and Urba'nski Page 1 x1. Introduction: Setting and Notation The purpose of this note is to demonstrate how the theory of infinite systems of conformal maps can be applied to obtain some results ab...
The pressure function for products of nonnegative matrices
 Math. Research Letter
"... Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure f ..."
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Cited by 7 (4 self)
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Abstract. Let (ΣA,σ) be a subshift of finite type and let M(x) be a continuous function on ΣA takingvalues in the set of nonnegative matrices. We extend the classical scalar pressure function to this new settingand prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where M(x) takes finite values M1, ·· ·,Mm. The pressure function reduces to P (q): = limn→ ∞ 1 log n J∈Σ ‖MJ ‖ A,n q. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps. 1.
Parabolic Iterated Function Systems
, 1999
"... In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, PerronFrobenius type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyp ..."
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Cited by 7 (6 self)
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In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, PerronFrobenius type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.
Rigidity Of MultiDimensional Conformal Iterated Function Systems
, 2000
"... The paper starts with an appropriate version of the bounded distortion theorem. We show that for a regular, satisfying the "Open Set Condition", iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space R d with d 3, the RadonNikodym derivati ..."
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Cited by 7 (4 self)
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The paper starts with an appropriate version of the bounded distortion theorem. We show that for a regular, satisfying the "Open Set Condition", iterated function system of countably many conformal contractions of an open connected subset of a Euclidean space R d with d 3, the RadonNikodym derivative d=dm has a realanalytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and is the unique probability invariant measure equivalent with m. Next, we explore in this context the concept of essential affinity of iterated function systems providing its several necessary and sufficient conditions. We prove the following rigidity result. If d 3 and h, a topological conjugacy between two not essentially affine systems F and G sends the conformal measure mF to a measure equivalent with the conformal measure mG , then h has a conformal extension on an open neighbourhood of the limit set of the system F . Finally in exactly the same way as in [MPU] we extend our rigidity result to the case of parabolic systems.
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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Cited by 7 (7 self)
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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.
Infinite Iterated Function Systems
, 1994
"... : We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the wellknown case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff d ..."
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Cited by 7 (3 self)
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: We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). We state results analogous to the wellknown case of finitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded nonempty sets not describable by IIFS. Keywords: Fractal geometry, iterated function systems, complete metric spaces, Baire space, Hausdorff measure, Hausdorff dimension, selfsimilarity. AMS classification: 28A80, 54E50, 54E52, 28A78, 54F45. 1. Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper [14], gained more and more interest. Several books on this topic are available [3, 7, 5, 18, 19] which have become popular even amo...