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We investigate matching for the family T α ( x ) = β x + α ( m o d 1 ) , α ∈ [ 0 , 1 ] , for fixed β > 1 . Matching refers to the property that there is an n ∈ N such that T α n ( 0 ) = T α n ( 1 ) . We show that for various Pisot numbers β , matching occurs on an open dense set of α ∈ [ 0 , 1 ] and we compute the Hausdorff dimension of its complement...
We use canonical Markov extensions (Hofbauer towers) to give an explicit construction of the natural extensions of various measure preserving endomorphisms, and present some applications to particular examples.
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