Tullio Ceccherini-Silberstein

Amenable groups and cellular automata

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Harmonic analysis on finite groups : representation theory, Gelfand pairs and Markov chains

Part I. Preliminaries, Examples and Motivations: 1. Finite Markov chains 2. Two basic examples on Abelian groups Part II. Representation Theory and Gelfand Pairs: 3. Basic representation theory of finite groups 4. Finite Gelfand pairs 5. Distance regular graphs and the Hamming scheme 6. The Johnson Scheme and the Laplace-Bernoulli diffusion model 7. The ultrametric space Part III. Advanced theory: 8. Posets and the q-analogs 9. Complements on representation theory 10. Basic representation theory of the symmetric group 11. The Gelfand Pair (S2n, S2 o Sn) and random matchings Appendix 1. The discrete trigonometric transforms Appendix 2. Solutions of the exercises Bibliography Index.

Growth and ergodicity of context-free languages

A language L over a finite alphabet E is called growth-sensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear context-free language is growth-sensitive. Ergodic means for a context-free grammar and language that its dependency di-graph is strongly connected. The same result as above holds for the larger class of essentially ergodic context-free languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for 2-block languages with a generating function technique regarding systems of algebraic equations.

Salem Numbers and Growth Series of Some Hyperbolic Graphs

Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs Xl,m associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick [FP94]) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We also prove that these denominators are essentially irreducible (they have a factor of X + 1 when m ≡ 2 mod 4; and when l = 3 and m ≡ 6 mod 12, for instance, they have a factor of X −X +1). We then derive some regularity properties for the coefficients an of the growth series: they satisfy Kλ − R < an < Kλ n + R for some constants K, R > 0, λ > 1.AbstractExtending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the ℓ-regular graphs

A generalization of the Curtis-Hedlund theorem

\mathcal{X}_{\ell ,m}

Context-free pairs of groups I

associated to regular tessellations of the hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We also prove that these denominators are essentially irreducible (they have a factor of X + 1 when m ≡ 2 mod 4; and when ℓ = 3 and m ≡ 4 mod 12, for instance, they have a factor of X2 − X + 1). We then derive some regularity properties for the coefficients fn of the growth series: they satisfy Kλn − R < fn < Kλn + R for some constants K, R < 0, λ < 1.

On the entropy of regular languages

For every infinite sequence ω = x 1 x 2 ? , with x i ? { 0 , 1 } , we construct an infinite 4-regular graph X ω . These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space { 0 , 1 } ∞ . We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs X ω have intermediate growth.

On the density of periodic configurations in strongly irreducible subshifts

Let A be a set and let G be a group, and equip A^G with its prodiscrete uniform structure. Let @t:A^G->A^G be a map. We prove that @t is a cellular automaton if and only if @t is uniformly continuous and G-equivariant. We also give an example showing that a continuous and G-equivariant map @t:A^G->A^G may fail to be a cellular automaton when the alphabet set A is infinite.