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Fully Orthonormal Polynomials: Dynamics

Below we show two dynamics plots for points x in SG. On the left we plot log|φk(x)| versus log(k). We plot the degrees of the polynomials k in log scale because for some of the points (i.e. index(x) = 1) the periodicity of the dynamics is best displayed in log scale. Also, note that we take the absolute value of the values of the OP, which removes some of the structure of the dynamics.

On the right we plot log|φk+1(x)| versus log|φk(x)|. Again, we take the absolute value of the values of the OP, which removes some of the structure of the dynamics. In the case of the Legendre polynomials, we saw that the dynamics of the points were attracted to a circle centered around the origin. Here however, the points seem to be repulsed from some arbitrary central point. Some of the dynamics are periodic, and all of them show overall exponential growth.

We group oints by level of approximation to SG. Note we only show for the first four approximations: V0, V1, V2, and V3.

Underneath the dynamics plots we explain the addressing system of the points.

Points in V0


Points in V1


Points in V2


Points in V3



Addressing System

For each point x in SG, we assign an index. For the dynamics, we only use points up to level m=3 in SG. The first three indices are the boundary points labeled in counterclockwise order. The next three indices are the contraction mappings of SG of oder m=1, in numerical order α =0,1,2. The points which have not yet been labeled are then given indices, again in counterclockwise order. This pattern is continued for each contraction mapping, in increasing order of m. The indices are displayed in the image to the right; click the image to enlarge.