Iteration of polynomials, functional equations, and fractal zeta functions Peter Grabner joint work with Gregory Derfel & Fritz Vogl Institut für Analysis und Computational Number Theory Graz University of Technology 10.12.2012 International Conference on Advances on Fractals and Related Topics, Hong-Kong Peter Grabner Iteration of polynomials. . .
Motivation For certain fractals, for instance the Sierpiński gasket and its higher dimensional analogues, the eigenfunctions and eigenvalues of the Laplace operator follow a “self-similar” pattern: the fractal is approximated by a sequence of graphs ( G n ) n ∈ N , which are connected by embeddings of the vertex sets ϕ n : V n → V n + 1 . ϕ n − 1 ϕ n +1 ϕ n +2 ϕ n G n G n +1 G n +2 The time rescaling factor λ is the fraction between the speed of the particle in G n and G n − 1 . Peter Grabner Iteration of polynomials. . .
These embeddings ϕ n correspond to a rational function ψ , which relates the probability generating function of the random walk on G n to the probability generating function of the random walk on G n + 1 . Y ( n +1) Y ( n ) m +1 T m +1 z �→ ψ ( z ) 1 1 4 4 1 1 4 4 1 4 1 1 1 1 1 1 4 4 4 4 4 4 1 1 4 4 Y ( n ) Y ( n +1) m T m The time rescaling factor is given by λ = E ( T m + 1 − T m ) = ψ ′ ( 1 ) . Peter Grabner Iteration of polynomials. . .
Spectral decimation The function ψ also relates the eigenvalues of the discrete Laplacians on G n and G n + 1 : every eigenvalue of ∆ n + 1 is a preimage under ψ of an eigenvalue of ∆ n . For the Laplacian on G , i.e. the limit of the rescaled discrete Laplacians ∆ n this means that every eigenvalue of ∆ can be written as λ m lim n →∞ λ n ψ − n ( z 0 ) , where z 0 is an eigenvalue of ∆ 0 . The multiplicities a µ of the eigenvalues depend only on m . More precisely, we need that the multiplicities of the eigenvalues have a rational generating function. Peter Grabner Iteration of polynomials. . .
Poincaré functions The equation giving the eigenvalues of the Laplacian motivates to study the solutions of the functional equation Φ( λ z ) = p (Φ( z )) , where 1 p ( z ) = ψ ( 1 / z ) , if p is a polynomial. For instance, this happens for the Sierpiński gaskets. Peter Grabner Iteration of polynomials. . .
Φ and the spectrum The spectrum of the Laplacian can then be described as Φ ( − 1 ) ( A ) for a finite set A . The value distribution of Φ therefore encodes the spectrum. Peter Grabner Iteration of polynomials. . .
The eigenvalue counting function � N ( x ) = a µ ∆ u = − µ u µ< x the trace of the heat kernel � a µ e − µ t = � P ( t ) = p t ( x , x ) d H ( x ) , G − ∆ u = µ u as well as the spectral zeta-function � a µ µ − s ζ ∆ ( s ) = ∆ u = − µ u can be related to Φ . Peter Grabner Iteration of polynomials. . .
The spectral zeta function The spectral zeta function ζ ∆ can be given in the form � � µ − s , R w ( λ s ) ζ ∆ ( s ) = w ∈ A Φ( − µ )= w µ � = 0 where R w is the rational function encoding the multiplicities of the eigenvalues. The analytic continuation of the functions � µ − s Φ( − µ )= w µ � = 0 can be obtained from the asymptotic behaviour of Φ at ∞ . Peter Grabner Iteration of polynomials. . .
The poles of ζ ∆ Im 4 πi log 5 The functions 2 πi � µ − s log 5 Φ( − µ )= w µ � = 0 have poles on the line ℜ s = log 5 2, which log 5 3 log 5 2 Re cancel in the sum forming ζ ∆ . This is a general phenomenon for fully symmetric − 2 πi fractals, as was shown recently by Stein- log 5 hurst and Teplyaev. − 4 πi log 5 Peter Grabner Iteration of polynomials. . .
Zero counting and the harmonic measure The function Φ has infinitely many zeros, which come in geometric progressions with factor λ by Φ( λ z ) = p (Φ( z )) . Let � N Φ ( r ) = 1 | z | < r Φ( z )= 0 denote the zero counting function. Then the following are equivalent r →∞ r − ρ N Φ ( r ) exists lim t → 0 t − ρ µ ( B ( 0 , t )) exists. lim Peter Grabner Iteration of polynomials. . .
Applications The existence of an analytic continuation of ζ ∆ to the whole complex plane allows for the definition and computation of an according Casimir energy: Consider the operator P = − ∂ 2 ∂τ 2 − ∆ on ( R / 1 β Z ) × G , where β = 1 / ( kT ) . The eigenvalues of P are then given by 4 k 2 π 2 + λ n . β 2 Peter Grabner Iteration of polynomials. . .
Zeta function of P The zeta function of P is then given by � ∞ − 4 n 2 π 2 1 t t s − 1 dt . � ζ P ( s ) = K ( t ) e β 2 Γ( s ) 0 n ∈ Z Using the theta relation − 4 π 2 n 2 β t = e − β 2 n 2 � � β 2 e 2 √ π t 4 t n ∈ Z n ∈ Z we obtain β � s − 1 � � s − 1 � ζ P ( s ) = 2 √ π Γ( s )Γ ζ ∆ 2 2 � ∞ ∞ β e − β 2 n 2 4 t t s − 3 � 2 dt + √ π Γ( s ) K ( t ) 0 n = 1 Peter Grabner Iteration of polynomials. . .
Regularised determinant of P The regularised determinant (“product of eigenvalues”) of P is given by − ζ ′ � � det ( P ) = exp P ( 0 ) . From the expression obtained before, we get � ∞ ∞ ∞ � − 1 � + β e − β 2 n 2 − λ j t t − 3 ζ ′ � � 2 dt . P ( 0 ) = − βζ ∆ √ π 4 t 2 0 n = 1 j = 1 The integral and the summation over n can be evaluated explicitly, which gives ∞ 1 − e − β √ � − 1 � � � ζ ′ � λ j P ( 0 ) = − βζ ∆ − 2 ln . 2 j = 1 Peter Grabner Iteration of polynomials. . .
Casimir energy The energy of the system is then given by ∞ � λ j E = − 1 ∂ P ( 0 ) = 1 � − 1 � ∂β ζ ′ � 2 ζ ∆ + e β √ . 2 2 λ j − 1 j = 1 The zero point or Casimir energy of the system is then obtained by letting the temperature tend to 0, which is equivalent to letting β tend to ∞ . This gives E Cas = 1 � − 1 � 2 ζ ∆ . 2 Peter Grabner Iteration of polynomials. . .
Numerical computations The very explicit procedure used for the analytic continuation of ζ ∆ allows for the numerical computation of ζ ∆ ( − 1 2 ) to arbitrary precision. We computed E D Cas = 0 . 5474693544 . . . for the Casimir energy of the two-dimensional Sierpiński gasket with Dirichlet boundary conditions. E N Cas = 2 . 134394089264 . . . for Neumann boundary conditions. Peter Grabner Iteration of polynomials. . .
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