For systems presented asthe Vershik map on a simple ordered Bratteli diagram, flow equivalence correspondsto the relation generated by conjugacy and making a finite number of changes tothe Bratteli diagram (see ). While conjugate systems are flow equivalent, flow equivalence is typicallya weaker relation than conjugacy for example, any system ( X, T ) is flow equivalentto the return action on a discrete cross section of ( X, T ), ( Y, S ) are said to be flow equivalent if there is a homeo-morphism h : Σ T X → Σ S Y such that h preserves the orientation of the respective R -orbits. mapping class group, minimal subshift, suspension, flow equivalence. X, T ) is conjugate to a subshift of low complexity. Apart from some general analysis, we focus especially on the casewhen ( The purpose of this paper is to study the group M ( T ) when ( X, T ) is a minimalCantor system. Maps f, g ∈ Homeo + Σ T X are isotopic if they are connected by a path inHomeo + Σ T X, and we define the mapping class group M ( T ) to be the group ofisotopy classes of maps in Homeo + Σ T X. The sus-pension Σ T X (sometimes called the mapping torus) comes with a natural R -actionwhose orbits correspond to the T -orbits in X, and we let Homeo + Σ T X denote thetopological group of homeomorphisms of Σ T X which respect the orientation of theflow. X, T ) we mean a homeomorphism T : X → X where X isa Cantor set when T is minimal we call ( X, T ) a minimal Cantor system. M ( T ) and the Picard group of C ( X ) ⋊ T Z Introduction The mapping class group of a subshift of linear complexity 207. The mapping class group of a substitution system 156. The mapping class group of a Cantor system 33. We also show that when ( X, T ) is minimal, M ( T ) embedsinto the Picard group of the crossed product algebra C ( X ) ⋊ T Z. X, T ) is a minimal subshift oflinear complexity satisfying a no-infinitesimals condition, we show that M ( T )is virtually abelian. X, T ) is a subshift associated to asubstitution, the group M ( T ) is an extension of Z by a finite group for alarge class of substitutions including Pisot type, this finite group is a quotientof the automorphism group of ( X, T ). We study M ( T ), focusing on the case when ( X, T ) isa minimal subshift. The group M ( T ) can be inter-preted as the symmetry group of the system ( X, T ) with respect to the flowequivalence relation. O c t THE MAPPING CLASS GROUP OF A MINIMAL SUBSHIFTįor a homeomorphism T : X → X of a Cantor set X, the map-ping class group M ( T ) is the group of isotopy classes of orientation-preservingself-homeomorphisms of the suspension Σ T X.
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