Roy L. Adler

Algorithms for sliding block codes---An application of symbolic dynamics to information theory

Ideas which have origins in Shannons work in information theory have arisen independently in a mathematical discipline called symbolic dynamics. These ideas have been refined and developed in recent years to a point where they yield general algorithms for constructing practical coding schemes with engineering applications. In this work we prove an extension of a coding theorem of Marcus and trace a line of mathematics from abstract topological dynamics to concrete logic network diagrams.

Equivalence of topological Markov shifts

We show that any two topological Markov shifts both of whose topological entropy equals logn (for somen) are equivalent by a finitistic coding.

Geodesic flows, interval maps, and symbolic dynamics

Geodesic flows and interval maps are two topics in the theory of dynamical systems with a long mathematical history. The first of these seems to have originated with Jacobi who related the flows to the study of Hamiltonian systems (for a detailed description of the connection, see [CFS]). The second arises in diverse settings, such as the modelling of population genetics [Ma] and the frequency count of digits in continued fraction expansions [Bi]. In both subjects the main problem is to describe the distribution of orbits. Thus we wish to know how the geodesies spread over the manifold containing them and how iterates of points under an interval map vary over the interval. Ergodic theory provides answers to these questions, particularly the notions of ergodicity and invariant measure which will be elaborated below. At first sight the two topics seem unrelated, geodesic flow being a continuous time action and interval map a discrete one. Nevertheless, we shall relate them and their associated symbolic dynamics when the flow takes place on a compact surface of constant negative curvature. In this case we use a graphic approach enabling us to find a series of reductions from geodesic flow to interval map. In relating the topics, we show how each sheds light on the other. We use ergodicity of interval maps to prove ergodicity of flows and conversely. Furthermore, explicit formulas for invariant measures of interval maps can be derived from invariant measures for flows. This fact is interesting as there is a paucity of explicit formulas for invariant measures of interval maps. The steps in our reduction scheme are known to exist abstractly. However, for the dynamical systems considered here, the reductions are carried out by means of elementary geometry. Our graphic

Symbolic dynamics and Markov partitions

The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research.

The mathematics of halftoning

This paper describes some mathematical aspects of halftoning in digital printing. Halftoning is the technique of rendering a continuous range of colors using only a few discrete ones. There are two major classes of methods: dithering and error diffusion. Some discussion is presented concerning the method of dithering, but the main emphasis is on error diffusion.

The ergodic infinite measure preserving transformation of boole

G. Boole proved that the transformation φ of the real line, defined by φ(x)=x−1/x, preserves Lebesgue measure. A general method is applied to proving that φ is ergodic. Some further applications of the method are also indicated.

A construction of a normal number for the continued fraction transformation

Abstract It is shown that if the continued fractions of the rationals 1 2 , 1 3 , 2 3 , 1 4 , 2 4 , 3 4 , 1 5 , 2 5 , 3 5 , 4 5 ,… are concatenated, a normal continued fraction is obtained.

State splitting for variable-length graphs (Corresp.)

The state splitting algorithm of Adler, Coppersmith, and Hassner for graphs with edges of fixed length is extended to graphs with edges of variable lengths. This has the potential to improve modulation code construction techniques. Although the ideas of state splitting come from dynamical systems, completely graph-theoretic terms are used.

Topological conjugacy of linear endomorphisms of the 2-torus

We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincare.

Skew products of Bernoulli shifts with rotations. II

IfT is a weakly mixing skew product transformation defined byT(x, y)=σx, y+f(x) (mod 1)), where σ is a Bernoulli shift andf is a function satisfying a Hölder type condition and measurable with respect to the past of an independent partition of σ, thenT is Bernoulli.