R. Daniel Mauldin

Multifractal decompositions of Moran fractals

We present a rigorous construction and generalization of the multifractal decomposition for Moran fractals with infinite product measure. The generalization is specified by a system of nonnegative weights in the partition sum. All the usual (smooth) properties of the f(α) theory are recovered for the case that the weights are equal to unity. The generalized spectrum, f(α, w), is invariant to a group of gauge transformations of the weights, and, in addition, need no longer be concave. In case the fractal is a Cantor set generated by an iterated function system of similarities, α is the pointwise dimension of the measure. We discuss properties of some examples.

On the Hausdorff dimension of some graphs

Consider the functions Wb(x)= b-cn[1(bnX + On)--1(0n)] n=-oo where b > 1, 0 0 such that if b is large enough, then the Hausdorff dimension of the graph of Wb is bounded below by 2a (C/ ln b). We also show that if a function f is convex Lipschitz of order a, then the graph of f has a-finite measure with respect to Hausdorffs measure in dimension 2 a. The convex Lipschitz functions of order a include Zygmunds class A,. Our analysis shows that the graph of the classical van der WaerdenTagaki nowhere differentiable function has a-finite measure with respect to h(t)-=t/ In(1/t). We consider the Hausdorff dimension of the graphs of various continuous functions. We introduce a new geometric property of a function: convex Lipschitz of some order, and obtain an upper bound on the dimension of a graph with this property. In particular, our analysis includes functions of the form

Gibbs states on the symbolic space over an infinite alphabet

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Hölder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.

Measure and dimension functions: measurability and densities

During the past several years a new type of geometric measure and dimension have been introduced the packing measure and dimension see Su Tr and TT These notions are playing an increasingly prevalent role in various aspects of dy namics and measure theory Packing measure is a sort of dual of Hausdor measure in that it is de ned in terms of packings rather than coverings However in contrast to Hausdor measure the usual de nition of packing measure requires two limiting procedures rst the construction of a premeasure and then a second standard limit ing process to obtain the measure This makes packing measure somewhat delicate to deal with The question arises as to whether there is some simpler method for de ning packing measure and dimension In this paper we nd a basic limitation on this possibility We do this by determining the descriptive set theoretic com plexity of the packing functions Whereas the Hausdor dimension function on the space of compact sets is Borel measurable the packing dimension function is not On the other hand we show that the packing measure and dimension functions are measurable with respect to the algebra generated by the analytic sets Thus the usual sorts of measurability properties used in connection with Hausdor measure e g measures of sections and projections remain true for packing measure We now give a somewhat more detailed description of our results and we intro duce some notation Throughout this paper X d will be a Polish space that is a complete separable metric space We equip the space K X of non empty compact subsets of X with the Hausdor distance

Graph directed Markov systems on Hilbert spaces

We deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowens formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by�product of the mainstream of our investigations we prove a 4r�covering theorem for all metric spaces. It enables us to establish appropriate co�Frostman type theorems.

Baire functions, borel sets, and ordinary function systems

Ifis a family of real-valued functions defined on a set X, then there is a smallest family, B(qt), of real-valued functions defined on X which contains ~ and which is closed under the process of taking pointwise limits of sequences from B(~). This family is called the Baire system generated by cb. One method of generating B(~) from q5 is by iteration of the operation of pointwise limits of sequences: Let ~0 be ~ and for each ordinal c~ > 0, let q5 be the family of all pointwise limits of sequences from (J~<~ q)~. Then q5~1 -~ ~1+1 = B(qt), where ~o 1 is the first un- countable ordinal. This system was described by Ren6 Baire in this th6se, published in 1899 (2). This paper is meant to be an exposition of some of the main results concerning this process that have been obtained since then. The second section concerns itself with some properties of the classes ~, under the assumption that the family ~ forms a lattice. In the third section, a development of the relationship between the classes q~ and Borel type sets which are inverse image sets of functions in q~ is given. That section concludes with Hausdorffs notion of an ordinary function system. These systems are completely characterized by their inverse image sets and yield appealing coextensive processes of generating the Baire system and of generating a certain Borel system (a-algebra) of sets. The Baire order of a family of functions # is the first ordinal ~ such that ~ ~ ~+1 • The Baire order problem for C(X), the space of real- valued continuous functions on a topological space X, is studied in the fourth section. Two proofs, due to Lebesgue, are given to show that the Baire order of C(0, 1) is oJ 1 . Later in this section it is shown that the Baire order of C(X), with X compact and Te, distinguishes those spaces which contain perfect sets from those which do not (dispersed spaces); the Baire order of the dispersed spaces being 0 or I and the others ~o 1 . In the last section the Baire order of various families of functions 418

AnL 1 counting problem in ergodic theory

We give a negative solution to the following counting problem for measure preserving transformations. Forf∈L+1(μ), is it true that supn (Nn(f)(x)/n) <∞, μ a.e., where Nn(f)(x)=≠{k:f(Tkx)/k>1/n}? One of the consequences is the nonvalidity of J. Bourgain’s Return Time Theorem for pairs of (L1,L1) functions.

Inductive definability: Measure and category

The purpose of this paper is threefold. Our first purpose is to exposit a major concept from descriptive set theory, inductive definability, and to present some of the major results concerning inductive operators. The classical version of this theory is carried out in the first two sections and the effective version in the fifth section. Our second purpose is to demonstrate a powerful unity of viewpoint provided by inductive definitions. This is shown by deriving several wellknown results from this viewpoint which had been previously proved by diverse methods. This point is demonstrated in the first section by deriving several well-known examples of analytic (and coanalytic) sets which are not Bore1 sets; in the third and fourth sections by the proofs of some “faithful extension” and reflection theorems. Again, this point is demonstrated in the fifth section in the presentation of several results from effective descriptive set theory. Our third purpose is to present some new results. Our first new result is given in the third section where it is shown (Theorem 3.3) that several classical “definability” results may be unified with the use of inductive definitions. New results are given in Theorems 4.1 and 4.2 where we demonstrate a definability and reflection principle with respect to conditional probability distributions. In the fifth section, we present new proofs of several known results and give a new characterization of p(x), the least ordinal not recursive in x. 55 OOOl-8708/80/100055-36

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