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280
Iterated random functions
 SIAM Review
, 1999
"... Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys ..."
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Cited by 133 (1 self)
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Abstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average. 1. Introduction. The
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 82 (23 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Dynamical systems, Measures and Fractals via Domain Theory
 Information and Computation
, 1995
"... We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L ar ..."
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Cited by 68 (19 self)
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We introduce domain theory in dynamical systems, iterated function systems (fractals) and measure theory. For a discrete dynamical system given by the action of a continuous map f:X X on a metric space X, we study the extended dynamical systems (l/X,l/f), (UX, U f) and (LX, Lf) where 1/, U and L are respectively the Vietoris hyperspace, the upper hyperspace and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, U f). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be omegacontinuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of nondeterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.
Domain Theory and Integration
 Theoretical Computer Science
, 1995
"... We present a domaintheoretic framework for measure theory and integration of bounded realvalued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilis ..."
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Cited by 57 (12 self)
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We present a domaintheoretic framework for measure theory and integration of bounded realvalued functions with respect to bounded Borel measures on compact metric spaces. The set of normalised Borel measures of the metric space can be embedded into the maximal elements of the normalised probabilistic power domain of its upper space. Any bounded Borel measure on the compact metric space can then be obtained as the least upper bound of an !chain of linear combinations of point valuations (simple valuations) on the upper space, thus providing a constructive setup for these measures. We use this setting to define a new notion of integral of a bounded realvalued function with respect to a bounded Borel measure on a compact metric space. By using an !chain of simple valuations, whose lub is the given Borel measure, we can then obtain increasingly better approximations to the value of the integral, similar to the way the Riemann integral is obtained in calculus by using step functions. ...
The Metric Analogue of Weak Bisimulation for Probabilistic Processes
, 2002
"... We observe that equivalence is not a robust concept in the presence of numerical information  such as probabilities  in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization ..."
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Cited by 51 (2 self)
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We observe that equivalence is not a robust concept in the presence of numerical information  such as probabilities  in the model. We develop a metric analogue of weak bisimulation in the spirit of our earlier work on metric analogues for strong bisimulation. We give a fixed point characterization of the metric. This makes available coinductive reasoning principles and allows us to prove metric analogues of the usual algebraic laws for process combinators. We also show that quantitative properties of interest are continuous with respect to the metric, which says that if two processes are close in the metric then observable quantitative properties of interest are indeed close. As an important example of this we show that nearby processes have nearby channel capacities  a quantitative measure of their propensity to leak information.
Domains for Computation in Mathematics, Physics and Exact Real Arithmetic
 Bulletin of Symbolic Logic
, 1997
"... We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability dist ..."
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Cited by 48 (10 self)
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We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...
Brownian motion and harmonic analysis on Sierpinski carpets
 MR MR1701339 (2000i:60083
, 1999
"... Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to con ..."
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Cited by 45 (9 self)
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Abstract. We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting. 1
Absolutely continuous Bernoulli convolutions  a simple proof
 Math. Research Letters
, 1996
"... Abstract. The distribution νλ of the random series ∑ ±λn has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urbański, and Ledrappier showed the importance of these distributions in several problems in dynamical systems ..."
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Cited by 45 (10 self)
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Abstract. The distribution νλ of the random series ∑ ±λn has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urbański, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Garsia in 1962, that νλ is absolutely continuous for a.e. λ ∈ (1/2, 1). Here we give a considerably simplified proof of this theorem, using differentiation of measures instead of Fourier transform methods. This technique is better suited to analyze more general random power series.
Metrics for Labelled Markov Systems
, 2001
"... The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of ..."
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Cited by 42 (8 self)
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The notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions. Thus, a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent. This instability is due to the quantitative nature of probabilistic processes. In a situation where the process behaviour has a quantitative aspect there should be a more robust approach to process equivalence. This paper studies a metric between labelled Markov processes. This metric has the property that processes are at zero distance if and only if they are bisimilar. The metric is inspired by earlier work on logics for characterizing bisimulation and is related, in spirit, to the Hutchinson metric.
A Computational Model for Metric Spaces
 Theoretical Computer Science
, 1995
"... For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !continuous iff X is separable. The computational model BX is used to give domaintheoretic pro ..."
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Cited by 42 (8 self)
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For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !continuous iff X is separable. The computational model BX is used to give domaintheoretic proofs of Banach's fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique nonempty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an !continuous computational model for measure theory on a separable complete metric space X . 1 Introduction In this paper, we establish new connections between the theory of metric spaces and domain theory, the two basic mathematical structures in computer science. For every metric space X, we define a continuous poset (not necessar...