We prove the following analogues of the Lebesgue density theorem for two types of fractal subsets of U: cookie-cutter Cantor sets and the zero set of a Brownian path. Write C for the set, and jit for the positive finite Hausdorff measure on C. Then there exists a constant c (depending on the set C) such that for /x-almost every xeC,is the e-ball around x and d is the Hausdorff dimension of C. We also define analogues of Hausdorff dimension and Lebesgue density for subsets of the integers, and prove that a typical zero set of the simple random walk has dimension \ and density V(2/;r). a set of finite measure in a class of infinite ergodic measure-preserving transformations; this is joint work with J. Aaronson and M. Denker and will appear elsewhere. The theorem proved there is closely related to Chung and Erdos' work although the proof and interpretation are quite different.The purposes of the present paper are several. Firstly we introduce the notion of order-two density and develop its basic properties: consistency with respect to usual density (which however diverges almost surely for all the examples mentioned above); a Radon-Nikodym-like result for absolutely continuous measures; and the comparison with a hierarchy of order-n densities based on the Hardy-Riesz log averages (see [15]) and on the averaging operators of [14]. Secondly we introduce the techniques needed to prove existence of the order-two density for the examples mentioned above; the existence of order-2 density for the Hausdorff measure on these sets can be considered to be an analogue (for Hausdorff measure) of the Lebesgue density theorem. Finally, we want to show that there is a deep underlying connection between all of the techniques we use-even though they may at first seem disparate. The middle-third Cantor set is dealt with in some detail because it is possible there to show these connections. The analogies that one sees between the different situations are not precise but seem to be very helpful in suggesting problems and methods.The notion of order-two density is related to Mandelbrot's concept of lacunarity (see [24], especially pp. 315-318, for an intuitive description and illustrations). The lacunarity of a fractal should describe the degree to which the structure is fractured; one wants a way of comparing different sets of the same dimension or related sets of different dimensions. Order-two density provides a possible tool for making such comparisons. In the physics literature Smith, Fournier and Spiegel [33] observe that estimates of fractal dimension (they consider in particular the correlation dimension) can show log-oscillatory behaviour. When such oscillations occur, this brings added difficulties to the problem of numerical estimation of dimension. Smith, Fournier and Spiegel are suggesting that one can however make use of this oscillation as a way of measuring the 'textural property of fractal objects that Mandelbrot calls lacunarity'. But as they point out, if the sets are not strictly self-similar then in general the oscil...

Given a C 1+γ hyperbolic Cantor set C, we study the sequence Cn,x of Cantor subsets which nest down toward a point x in C. We show that Cn,x is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on D. Sullivan's dual Cantor set. We show the limit sets are themselves C k+γ , C ∞ or C ω hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the C 1+γ conjugacy class of C. The proof of this leads to the following rigidity theorem: if two C k+γ , C ∞ or C ω hyperbolic Cantor sets are C 1 -conjugate, then the conjugacy (with a different extension) is in fact already C k+γ , C ∞ or C ω . Within one C 1+γ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Conjugacy classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the C 1 norm.Typeset by A M S-T E X 1

Abstract. We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

We analyse the correlation and limit behaviour of partial sums for the stationary stochastic process (f (T t (x)), µ), t = 0, 1,. .. , for functions f of superpolynomial variation, the class SP defined below (which includes the Hölder functions), where T : + → + is the left shift map on + = ∞ 0 {0, 1} and µ is the non-atomic equilibrium measure of a non-Hölder potential g = g γ belonging to a one-parameter family, indexed by γ > 2. First, using the renewal equation, we show a polynomial rate of convergence for the associated Ruelle operator for cylinder set observables. We then use these estimates to prove the following theorems:

Abstract. We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. This is a sequence of diffeomorphisms along compact Riemannian manifolds such that the tangent bundles split into expanding and contracting subspaces.We develop the general theory, studying sequences of maps up to a notion of isomorphism and with respect to an equivalence relation generated by two natural operations, gathering and dispersal.Then we concentrate on linear Anosov families on the two-torus. We study in detail a basic class of examples, the multiplicative families, and a canonical dispersal of these, the additive families. These form a natural completion to the collection of all linear Anosov maps.A renormalization procedure constructs a sequence of Markov partitions consisting of two rectangles for a given additive family. This codes the family by the nonstationary subshift of finite type determined by exactly the same sequence of matrices.Any linear positive Anosov family on the torus has a dispersal which is an additive family. The additive coding then yields a combinatorial model for the linear family, by telescoping the additive Bratteli diagram. The resulting combinatorial space is then determined by the same sequence of nonnegative matrices, as a nonstationary edge shift. This generalizes and provides a new proof for theorems of Adler and Manning.