/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable.

Universality and complexity in cellular automata

We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.

Non-archimedean amoebas and tropical varieties

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group LetR

 d
 = ℤ[u

 1
 ±1
 ,,[d

 1
 ±1
 ] be the ring of Laurent polynomials ind commuting variables, andM be anR

 d
 -module Then the dual groupX

 M
 ofM is compact, and multiplication onM by each of thed variables corresponds to an action α
 M
 of ℤ
 d
 by automorphisms ofX

 M
  Every action of ℤ
 d
 by automorphisms of a compact abelian group arises this way Iff∋R

 d
 , our main formula shows that the topological entropy of
 
$$\alpha _{R_d /\left\langle f \right\rangle } $$

 is given by
 
$$h\left( {\alpha _{R_d /\left\langle f \right\rangle } } \right) = \log M\left( f \right) = \int\limits_0^1 \ldots \int\limits_0^1 {\log \left| {f\left( {e^{2\pi it_1 } , \ldots , e^{2\pi it_d } } \right)} \right|} dt_1 \ldots dt_d $$

 , where M(f) is the Mahler measure off This reduces to the classical result for toral automorphisms via Jensen's formula While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7ζ(3)/4π2 occur If p is a non-principal prime ideal, we show
 
$$h\left( {\alpha _{R_d /p} } \right) = 0$$

  Using an analogue of the Yuzvinskii-Thomas addition formula, we computeh(α
 M
 ) for arbitraryR

 d
 -modulesM, and then the joint entropy for an action of ℤ
 d
 on a (not necessarily abelian) compact group Using a result of Boyd, we characterize those α
 M
 which have completely positive entropy in terms of the prime ideals associated toM, and show this condition implies that α
 M
 is mixing of all orders We also establish an analogue of Berg's theorem, proving that if α
 M
 has finite entropy then Haar measure is the unique measure of maximal entropy if and only if α
 M
 has completely positive entropy Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

/pdf/mahler-measure-and-entropy-for-commuting-automorphisms-of-4x9qv1zcx2.pdf

Mahler measure and entropy for commuting automorphisms of compact groups.

We give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.

/pdf/the-entropies-of-topological-markov-shifts-and-a-related-1qimeift6k.pdf

The entropies of topological Markov shifts and a related class of algebraic integers

Let (XT,AT) be a shift of finite type, and G = aut(vT) denote the group of homeomorphisms of XT commuting with ¢T. We investigate the algebraic properties of the countable group G and the dynamics of its action on XT and associated spaces. Using "marker" constructions, we show G contains many groups, such as the free group on two generators. However, G is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on n coordinates leads to a new and nontrivial topological invariant of CRT whose exact value is not known. We prove that, modulo a few points of low period, G acts transitively on the set of points with least aT-period n. Using padic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of G on the dimension group of aT iS investigated. We show there are no proper infinite compact G-invariant sets. We give a complete characterization of the G-orbit closure of a continuous probability measure, and deduce that the only continuous G-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.

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Douglas Lind

Papers

An Introduction to Symbolic Dynamics and Coding

Non-archimedean amoebas and tropical varieties

Mahler measure and entropy for commuting automorphisms of compact groups.

The entropies of topological Markov shifts and a related class of algebraic integers

The automorphism group of a shift of finite type