On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

Table 2

Substitution rules most widely considered in the study of self-similar systems, where 𝑛 and 𝑚 are positive integers.

Sequence	Set 𝒜	Substitution rule
Fibonacci	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵; 𝑔(𝐵)=𝐴
Silver mean	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐴𝐵; 𝑔(𝐵)=𝐴
Bronze mean	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐴𝐴𝐵; 𝑔(𝐵)=𝐴
Precious means	{𝐴,𝐵}	𝑔(𝐴)=𝐴^𝑛𝐵; 𝑔(𝐵)=𝐴
Copper mean	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵𝐵; 𝑔(𝐵)=𝐴
Nickel mean	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵𝐵𝐵; 𝑔(𝐵)=𝐴
Metallic means	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵^𝑛; 𝑔(𝐵)=𝐴
Mixed means	{𝐴,𝐵}	𝑔(𝐴)=𝐴^𝑛𝐵^𝑚; 𝑔(𝐵)=𝐴
Thue-Morse	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵; 𝑔(𝐵)=𝐵𝐴
Period-doubling	{𝐴,𝐵}	𝑔(𝐴)=𝐴𝐵; 𝑔(𝐵)=𝐴𝐴
Ternary Fibonacci	{𝐴,𝐵,𝐶}	𝑔(𝐴)=𝐴𝐶; 𝑔(𝐵)=𝐴; 𝑔(𝐶)=𝐵
Rudin-Shapiro	{𝐴,𝐵,𝐶,𝐷}	𝑔(𝐴)=𝐴𝐵; 𝑔(𝐵)=𝐴𝐶; 𝑔(𝐶)=𝐷𝐵; 𝑔(𝐷)=𝐷𝐶
Paper folding	{𝐴,𝐵,𝐶,𝐷}	𝑔(𝐴)=𝐴𝐵; 𝑔(𝐵)=𝐶𝐵; 𝑔(𝐶)=𝐴𝐷; 𝑔(𝐷)=𝐶𝐷