Canonical symbolic dynamics for one-dimensional generalized solenoids

Inhyeop Yi
2001 Transactions of the American Mathematical Society  
We define canonical subshift of finite type covers for Williams' onedimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids. 3741 3742 INHYEOP YI Apart from the matter of canonical symbolic dynamics, we mention renewed interest in Williams' systems and related systems on account of connections with ordered group invariants ([3, 12, 18, 19]) and substitutions and tilings ([2, 6]). We study the 1-solenoid systems as purely topological systems.
more » ... For this we give some defining topological axioms closely related to Williams' axioms. A 1solenoid of Williams becomes one of our 1-solenoids by ignoring the differentiable structure. Conversely, every topologically defined 1-solenoid can be given a differentiable structure which makes it a 1-solenoid in the sense of Williams. However, the essential aspects of the situation are not differentiable but topological, and to clarify this we give the purely topological development. The outline of the paper is as follows. In Section 2, following Williams ([14, 15]) rather closely, we give axioms for our systems and prove some basic facts about them. We also recall the construction of an SFT cover from a graph presentation. In Section 3, we recall Williams' definition of shift equivalence and show that every topological conjugacy of shift maps on branched 1-solenoids is induced by a shift equivalence of their graph presentations. (A 'branched' solenoid is a solenoid derived from a presentation which need not satisfy Williams' Flattening Axiom, so this is a slight generalization of Williams' work.) We also establish a key observation: If the shift equivalence is given by graph maps (maps sending vertices to vertices), then the conjugacy lifts uniquely to a conjugacy of the SFT covers derived from the graph presentations. In Section 4, given O, we give a graph algorithm for a new graphical presentation (X O , f O ) of the solenoid system. One consequence of this construction is that every 1-solenoid system with a fixed point has an elementary presentation in the sense of Williams, so this extends Williams' classification result ([15, §7]) to all 1-solenoid systems with fixed points. In Section 5, we use the previous results to produce the canonical SFT covers, and use them to distinguish the pair of systems considered by Williams and Ustinov ([13, 15] ) by computing Bowen-Franks groups of certain attached canonical SFT covers. In Appendix A, we show that our canonical SFT covers are not canonical as one-sided SFTs, despite the one-sided aspects of the construction. In Appendix B, we show that our graph presentations of topological 1-solenoid systems can be given a differentiable structure making them branched manifolds with differentiable immersions in the sense of Williams. Markov maps and their SFT covers In the style of Williams ([14, 15]), we will define several axioms which might be satisfied by a continuous self-map of a directed graph. Let X be a directed graph with vertex set V and edge set E, and f : X → X a continuous map. Axioms 0-3 and 5 correspond to Williams' Axioms 0-2, 3 • , and 4 in [15] .
doi:10.1090/s0002-9947-01-02710-6 fatcat:6of37w2agngptorweqzbu5dr6i