Results 1  10
of
70
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 ..."
Abstract

Cited by 30 (9 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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
"... ..."
Geometric Thermodynamical Formalism and Real Analyticity for Meromorphic Functions of Finite Order
 Ergod. Th. & Dynam. Sys
"... Abstract. Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–roo ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
Abstract. Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved. 1.
Valuations of Languages, with Applications to Fractal Geometry
, 1994
"... Valuations  morphisms from (\Sigma ; \Delta; ) to ((0; 1); \Delta; 1)  are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [28, 41, 11, 9, 10, 42]. This paper shows that valuations are useful not only within the theory of codes, but also when dealing wit ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Valuations  morphisms from (\Sigma ; \Delta; ) to ((0; 1); \Delta; 1)  are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [28, 41, 11, 9, 10, 42]. This paper shows that valuations are useful not only within the theory of codes, but also when dealing with ambiguity, especially in contextfree grammars, or for defining outer measures on the space of omegawords which are of some importance to the theory of fractals. These connections yield new formulae to determine the Hausdorff dimension of fractal sets (especially in Euclidean spaces) defined via formal languages. The class of fractals describable with contextfree languages strictly includes that of MRFSfractals introduced in [58, 18]. Some of the results of this paper also appear as part of the author's PhD thesis [36] and in [30, 31].
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 ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
: 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...