In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

Local entropy theory

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Invariant measures on stationary Bratteli diagrams

We consider a hierarchy of notions of largeness for subsets of Z (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βZ to establish the connections between various notions of largeness and apply the obtained results to the study of sets of “fat intersections” R2 A,B = {n ∈ Z : μ(A ∩ T nB) > μ(A)μ(B) − 2}. Among other things we show that the sets R2 A,B allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary Z-actions.

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Large sets of integers and hierarchy of mixing properties of measure preserving systems

To evaluate the performance of randomization designs under various parameter settings and trial sample sizes, and identify optimal designs with respect to both treatment imbalance and allocation randomness, we evaluate 260 design scenarios from 14 randomization designs under 15 sample sizes range from 10 to 300, using three measures for imbalance and three measures for randomness. The maximum absolute imbalance and the correct guess (CG) probability are selected to assess the trade-off performance of each randomization design. As measured by the maximum absolute imbalance and the CG probability, we found that performances of the 14 randomization designs are located in a closed region with the upper boundary (worst case) given by Efron's biased coin design (BCD) and the lower boundary (best case) from the Soares and Wu's big stick design (BSD). Designs close to the lower boundary provide a smaller imbalance and a higher randomness than designs close to the upper boundary. Our research suggested that optimization of randomization design is possible based on quantified evaluation of imbalance and randomness. Based on the maximum imbalance and CG probability, the BSD, Chen's biased coin design with imbalance tolerance method, and Chen's Ehrenfest urn design perform better than popularly used permuted block design, EBCD, and Wei's urn design.

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Quantitative comparison of randomization designs in sequential clinical trials based on treatment balance and allocation randomness.

The demand for unequal allocation in clinical trials is growing. Most commonly, the unequal allocation is achieved through permuted block randomization. However, other allocation procedures might be required to better approximate the allocation ratio in small samples, reduce the selection bias in open-label studies, or balance on baseline covariates. When these allocation procedures are generalized to unequal allocation, special care is to be taken to preserve the allocation ratio at every allocation step. This paper offers a way to expand the biased coin randomization to unequal allocation that preserves the allocation ratio at every allocation. The suggested expansion works with biased coin randomization that balances only on treatment group totals and with covariate-adaptive procedures that use a random biased coin element at every allocation. Balancing properties of the allocation ratio preserving biased coin randomization and minimization are described through simulations. It is demonstrated that these procedures are asymptotically protected against the shift in the rerandomization distribution identified for some examples of minimization with 1:2 allocation. The asymptotic shift in the rerandomization distribution of the difference in treatment means for an arbitrary unequal allocation procedure is explicitly derived in the paper.

Preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation.

The Euler graph has vertices labelled (n, k) for n = 0, 1, 2, ... and k = 0, 1, ..., n, with k + 1 edges from (n, k) to (n + 1, k) and n − k + 1 edges from (n, k) to (n + 1, k + 1). The number of paths from (0,0) to (n, k) is the Eulerian number A(n, k), the number of permutations of 1,2,...,n+ 1 with exactly n− k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure. 1. The Euler Graph The Euler graph is an infinite directed graph such that at level n there are n + 1 vertices labelled (n, 0) through (n, n). The vertex (n, k) has n+ 2 total edges leaving it, with k + 1 edges connecting it to vertex (n+ 1, k) and n− k + 1 edges connecting it to vertex (n+ 1, k + 1). Define X to be the space of infinite edge paths on the Euler graph. X is a compact metric space in a natural way: if two paths x = x0x1x2... and y = y0y1y2... agree for all n less than j and xj 6= yj , then define d(x, y) = 2 −j . The number of paths from the root vertex (0,0) to the vertex (n, k) is the Eulerian number, A(n, k), which is the number of 2 S. Bailey, M. Keane, K. Petersen, and I. Salama

Ergodicity of the adic transformation on the Euler graph

Selection bias is a potential problem, especially in trials where blinding is not possible. Randomization with a constrained block of large size provides better protection from selection bias than using a sequence of blocks of small size with the same maximum imbalance. We propose an algorithm that efficiently generates constrained block allocation sequences, and we describe two clinical trials in which such a constrained randomization was used.

Efficient generation of constrained block allocation sequences.

Tree shift topological entropy

Preface Part I. Survey Articles: 1. Pointwise ergodic theorems via harmonic analysis Joseph M. Rosenblatt and Mate Wierdl 2. Harmonic analysis in rigidity theory R. J. Spatzier 3. Some properties and applications of joinings in ergodic theory J.-P. Thouvenot Part II. Research Papers: 4. Ergodic baker's transformations C. J. Bose and P. Grezegorczyk 5. Almost sure convergence of projections to self-adjoint operators in L2(0,1) Lech Ciach, Ryszard Jajte and Adam Paskievicz 6. Quasi-uniform limits of uniformly recurrent points Tomasz Downarowicz 7. Strictly nonpointwise Markov operators and weak mixing Tomasz Downarowicz 8. Two techniques in multiple recurrence A. H. Forrest 9. For Bernoulli transformations the smallest natural family of factors consists of all factors Eli Glasner 10. Topological entropy of extensions Eli Glasner and Benjamin Weiss 11. Functional equations associated with the spectral properties of compact group extensions Geoffrey Goodson 12. Multiple recurrence for nilpotent groups of affine transformations of the 2-torus Daniel A. Hendrick 13. A remark on isometric extensions in relatively independent joinings Emmanuel Lesigne 14. Three results in recurrence Randall McCutcheon 15. Calculation of the limit in the return times theorem for Dunford-Schwartz operators James H. Olsen 16. Eigenfunctions of T x S and the Conze-Lesigne algebra Daniel J. Rudolph Conference program List of participants.

Ergodic theory and its connections with harmonic analysis : proceedings of the 1993 Alexandria conference

Spearman's footrule, a well-known measure of rank correlation, cannot be standardized satisfactorily because it is not symmetric under reversal of one variate. We propose a symmetrized version, derive some of its properties, and provide tables for small samples.

Ibrahim Salama

Papers

Ergodicity of the adic transformation on the Euler graph

Efficient generation of constrained block allocation sequences.

Tree shift topological entropy

Ergodic theory and its connections with harmonic analysis : proceedings of the 1993 Alexandria conference

The symmetric footrule