Cutting sequence

Cutting sequence is a concept in the field of mathematics, particularly within the areas of geometry and dynamical systems. It is a sequence of symbols that represents the intersection of a straight line or curve with a set of parallel lines or a lattice in the plane. Cutting sequences are used to study the properties of these intersections and have applications in various mathematical disciplines, including ergodic theory, symbolic dynamics, and the study of quasicrystals.

Definition

A cutting sequence is generated by considering a straight line or curve in the plane and a set of parallel lines or a lattice. As the line or curve intersects the parallel lines or lattice, a sequence of symbols is produced, with each symbol representing a specific direction or type of intersection. The exact nature of the symbols and the rules for their generation depend on the geometry of the line or curve and the arrangement of the parallel lines or lattice.

Applications

Cutting sequences have a wide range of applications in mathematics:

• In ergodic theory, cutting sequences are used to study the statistical properties of geometric objects and their transformations.
• In symbolic dynamics, they provide a way to encode geometric information in a symbolic form, facilitating the study of dynamical systems through combinatorial methods.
• Cutting sequences also play a role in the study of quasicrystals, where they are used to understand the arrangement of atoms in materials that exhibit a quasiperiodic structure.

Examples

One of the simplest examples of a cutting sequence is generated by a straight line intersecting a set of parallel lines. If the line intersects the parallel lines at a constant angle, the cutting sequence can be represented by a repeating sequence of symbols, each corresponding to an intersection with a line.

Another example involves a straight line intersecting a square lattice. In this case, the cutting sequence can include two symbols, one for intersections with horizontal lines and another for intersections with vertical lines. The sequence then reflects the path of the line through the lattice.

See Also

• Geodesic
• Symbolic dynamics
• Ergodic theory
• Quasicrystals