Selim Tuncel

Infinite-to-one codes and Markov measures

We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.

Iterative Decoding of Tail-Biting Trellises and Connections with Symbolic Dynamics

The sum-product and min-sum algorithms are used to decode codes defined by trellises. In this paper, we discuss the behavior of these and related algorithms on tail-biting (TB) trellises.

Surjective extensions of sliding-block codes

Several constructions are presented for extending a bounded-to-one sliding-block code to a bounded-to-one surjection onto its range, while preserving nice properties of the original code.

A Spanning Tree Invariant for Markov Shifts

We introduce a new type of invariant of block isomorphism for Markov shifts, defined by summing the weights of all spanning trees for a presentation of the Markov shift. We give two proofs of invariance. The first uses the Matrix-Tree Theorem to show that this invariant can be computed from a known invariant, the stochastic zeta function of the shift. The second uses directly the definition to show invariance under state splitting, from which all block isomorphisms can be built.

Handelman’s Theorem on Polynomials with Positive Multiples

For a polynomial p in several variables and a face F of its Newton polytope, let PF denote the polynomial consisting of the terms of P that lie in F , with the coefficients given by p. Handelman’s theorem states that p has a polynomial multiple with positive coefficients if and only if no PF has a zero with strictly positive coordinates. We give a short and self-contained account of its proof.

When does a polynomial ideal contain a positive polynomial

Abstract We use Grobner bases and a theorem of Handelman to show that an ideal I of R [x 1 ,…,x k ] contains a polynomial with positive coefficients if and only if no initial ideal inv(I), v∈ R k , has a positive zero.

Constructing finitary isomorphisms with finite expected coding times

This paper is motivated by the question of whether the invariants β, Δ,cΔ completely characterize isomorphism of Markov chains by finitary isomorphisms that have finite expected coding times (fect). We construct a finitary isomorphism with fect under an additional condition. Whether coincidence of β, Δ,cΔ implies the required condition remains open.

Coefficient rings for beta function classes of Markov chains

To each function we associate a period, a polytope, a group

Boundary measures of Markov chains

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On automorphisms of Markov chains

, a subgroup