Periodic Quasiperiodic Dielectric $\epsilon \left(x,y\right)=\sum _{{G}_{1},{G}_{2}}{\epsilon }_{{G}_{1},{G}_{2}}{\text{e}}^{i\left({G}_{1}x+{G}_{2}y\right)}$ $y=\tau x$ $\epsilon \left(x,\tau x\right)=\sum _{{G}_{1},{G}_{2}}{\epsilon }_{{G}_{1},{G}_{2}}{\text{e}}^{i\left({G}_{1}+{G}_{2}\tau \right)x}$ (Projection of Periodic Case) Eigenfunction Hz ${\text{e}}^{i\left({k}_{1}x+{k}_{2}y\right)}$ ${\text{e}}^{i\left({k}_{1}+{k}_{2}\tau \right)x}$ (Again, Projection of Periodic Case) Periodicity in Configuration/Physical Space Yes No Periodicity in Reciprocal Space Yes Yes Bloch Theorem Applies Geometric Periodicity =>Eigenfunction Periodicity Does Not Apply, (due to 3rd point above) Chaos and renormalization on the circle ${\text{e}}^{i{G}_{2}\cdot slope\left(x+a\right)}={\text{e}}^{i{G}_{2}\cdot slope\cdot x}$ , Requires: $\left({\text{e}}^{i{G}_{2}\cdot slope\cdot a}\right)={\left({\text{e}}^{i{G}_{2}a}\right)}^{slope}={\left(1\right)}^{slope}=1$ And the last step requires that slope = integer or rational (with adjusted periodicity) otherwise multivaluedness of roots of unity kicks in and in our case slope = τ = irrational But we still have a relation between Geometry and Physics(and in fact quasiperiodic periodic problem needs to be solved in conjunction with periodic problem two unknown frequencies require two equations see below): Projective Periodicity of Geometry => Projective Periodicity of Eigenfunction Temporal Dependence $H\left(x,y;t\right)=H\left(x,y\right){\text{e}}^{-i\omega t}=H\left(x,y\right){\text{e}}^{-i\left({\omega }_{1}+{\omega }_{2}\right)t}$ $H\left(x;t\right)=H\left(x\right){\text{e}}^{-i\left({\omega }_{1}+{\omega }_{2}\tau \right)t}$ Mechanical Analogy Isotropic Oscillator Or Anisotropic Oscillator (commensurate case) Anisotropic Oscillator(incommensurate case)