Dynamical properties of the horseshoe map

Key Points

• To demonstrate that the horseshoe map shares dynamical properties with the shift map using an alternative proof approach.
• Utilized the Cantor map to establish relationships between the horseshoe map and the shift map.
• Defined an equivalence relation on Cantor sets within the unit rectangle.
• Confirmed that the horseshoe map inherits the dynamical characteristics of the shift map through this new proof.
• Established a clear connection between the properties of both maps under the equivalence relation.