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

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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_{+}(B) of positive invariant tetrahedra of B. We show that T_{+}(B) has finitely many components. For many cases of B, we show that T_{+}(B) is path connected. We also give examples of B for which T_{+}(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_{+} (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_{+}, but for which there is no uniform bound on the lag and matrix sizes of the strong shift equivalences required.

For an SFT (X_{A},σ_{A}), let M_{A} denote the mapping class group of σ_{A}. M_{A} is the group of flow equivalences of the mapping torus Y_{A}, (i.e., self homeomorphisms of Y_{A} which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of Y_{A} isotopic to the identity. In the second chapter, we prove several results for the mapping class group M_{A} of a nontrivial irreducible SFT (X_{A},σ_{A}) as follows. For every n in N, M_{A} acts n-transitively on the set of circles in the mapping torus Y_{A} of (X_{A},σ_{A}). The center of M_{A} is trivial. M_{A} contains an embedded copy of Aut(σ_{B})/<σ_{B}> for any SFT (X_{B},σ_{B}) flow equivalent to (X_{A},σ_{A}). A flow equivalence F:Y_{A} → Y<subA 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_{A} (such a return map is an irreducible SFT flow equivalent to σ_{A}). However, there exist elements of M_{A} containing no flow equivalence with an invariant cross section. Finally, we define the groupoid PE_{Z}(A) of positive equivalences from A. There is an associated surjective group homomorphism π:PE_{Z}(A)→ M_{A}/S_{A} (where S_{A} is the normal subgroup of M_{A} 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_{Z}(A) → SL(Z). We show that for every [F] in M_{A}/S_{A} and V in SL(Z) there exists g in PE_{Z}(A) such that π_{A}(g) = [F] and ρ_{A}(g)=V.