Deformations of limit sets down parabolic pleating rays

A journey down the 1/2-pleating ray (i.e. the positive imaginary axis).

A journey down the 1/2-pleating ray (Markov algorithm).

A journey down the 1/3-pleating ray.

A journey down a curve given by a pertubation of a low-order term of the 1/3-slope Farey polynomial, showing what a trip down a ray which is a deformation of a pleating ray looks like.

A journey down the 4/5-pleating ray.

The branches of the hyperbolic locus of \(\Phi_{1/4}\)

The inverse images of \( t \) under the Farey polynomial of slope \( 1/4 \), with \( t \) now running forwards from \( -100 \) to \( 0 \). This exhibits the pleating ray as the branch of the hyperbolic
locus of the Farey polynomial which leaves the Riley slice last.

A journey down the 1/4-pleating ray (i.e. the branch of the locus shown in the previous video in the top-right quadrant).

A journey down the branch of the hyperbolic locus of \( \Phi_{1/4} \) with asymptotic slope \( 3\pi/4 \) (i.e. the branch of the locus in the top-left quadrant).

Deformations of limit sets down elliptic pleating rays

A journey down the 1/2-pleating ray for the group with cone angles \( 2\pi/3 \) and \( 2\pi/4 \) (Markov algorithm).

A journey down the 1/2-pleating ray for the group with cone angles \( 2\pi/4 \) and \( 2\pi/4 \) (Markov algorithm).

A journey down the 1/3-pleating ray for the group with cone angles \( 2\pi/3 \) and \( 2\pi/4 \).

A journey down the 1/3-pleating ray for the group with cone angles \( 2\pi/3 \) and \( 2\pi/4 \) (Markov algorithm).

A journey down the 1/3-pleating ray for the group with cone angles \( 2\pi/4 \) and \( 2\pi/4 \) (Markov algorithm).

Miscellaneous

The inverse images of \( t \) under the Farey polynomials of slope denominator \( \leq 32 \), as \( t \) decreases from 0 to \( -30 \). Observe that the pleating rays converge to the boundary at a rate
corresponding to the slope denominator.

The roots of \( \Phi_\alpha \) as \( \alpha \) runs from \( 0 \) to \( 1 \) over the rational numbers of denominator \( \leq 32 \).