Positive Rational Strong Shift Equivalence and The Mapping Class Group of A Shift of Finite Type

Positive Rational Strong Shift Equivalence and The Mapping Class Group of A Shift of Finite Type

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2011

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Chuysurichay, Sompong

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Boyle, Michael

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This thesis studies two independent topics in symbolic dynamics, the positive rational strong shift equivalence and the mapping class group of a shift of finite type.
In the first chapter, we give several results involving strong shift equivalence of positive matrices over the rational or real numbers, within the path component framework of Kim and Roush. Given a real matrix B with spectral radius less than 1, we consider the number of connected components of the space T<sub>+</sub>(B) of positive invariant tetrahedra of B. We show that T<sub>+</sub>(B) has finitely many components. For many cases of B, we show that T<sub>+</sub>(B) is path connected. We also give examples of B for which T<sub>+</sub>(B) has 2 components. If S is a subring of R containing Q we show that every primitive matrix over S with positive trace is strong shift equivalent to a positive doubly stochastic matrix over S<sub>+</sub> (and consequently the nonzero spectra of primitive stochastic positive trace matrices are all achieved by positive doubly stochastic matrices). We also exhibit a family of 2 by 2 similar positive stochastic matrices which are strong shift equivalent over R<sub>+</sub>, but for which there is no uniform bound on the lag and matrix sizes of the strong shift equivalences required.
For an SFT (X<sub>A</sub>,σ<sub>A</sub>), let M<sub>A</sub> denote the mapping class group of σ<sub>A</sub>. M<sub>A</sub> is the group of flow equivalences of the mapping torus Y<sub>A</sub>, (i.e., self homeomorphisms of Y<sub>A</sub> which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of Y<sub>A</sub> isotopic to the identity. In the second chapter, we prove several results for the mapping class group M<sub>A</sub> of a nontrivial irreducible SFT (X<sub>A</sub>,σ<sub>A</sub>) as follows. For every n in N, M<sub>A</sub> acts n-transitively on the set of circles in the mapping torus Y<sub>A</sub> of (X<sub>A</sub>,σ<sub>A</sub>). The center of M<sub>A</sub> is trivial. M<sub>A</sub> contains an embedded copy of Aut(σ<sub>B</sub>)/<σ<sub>B</sub>> for any SFT (X<sub>B</sub>,σ<sub>B</sub>) flow equivalent to (X<sub>A</sub>,σ<sub>A</sub>). A flow equivalence F:Y<sub>A</sub> → Y<subA</sub> has an invariant cross section if and only if F is induced by an automorphism of the first return map to some cross section of Y<sub>A</sub> (such a return map is an irreducible SFT flow equivalent to σ<sub>A</sub>). However, there exist elements of M<sub>A</sub> containing no flow equivalence with an invariant cross section. Finally, we define the groupoid PE<sub>Z</sub>(A) of positive equivalences from A. There is an associated surjective group homomorphism π:PE<sub>Z</sub>(A)→ M<sub>A</sub>/S<sub>A</sub> (where S<sub>A</sub> is the normal subgroup of M<sub>A</sub> generated by Nasu's simple automorphisms of return maps to cross sections). In the case of trivial Bowen-Franks group, there is another group homomorphism, ρ:PE<sub>Z</sub>(A) → SL(Z). We show that for every [F] in M<sub>A</sub>/S<sub>A</sub> and V in SL(Z) there exists g in PE<sub>Z</sub>(A) such that π<sub>A</sub>(g) = [F] and ρ<sub>A</sub>(g)=V.