Smale horseshoe via the anti-integrability

The embedding of the Bernoulli shift into the horseshoe map can be viewed as it is inherited from the anti-integrable limit where the horseshoe has infinite contraction in one direction and infinite expansion in the other. At the limit the map is virtually a subshift of finite type with four symbols or equivalently a full shift with two symbols. We present an algebraic explanation for the equivalence of the two shifts.