Ruelle operators: Functions which are harmonic with respect to a transfer operator (Q2734763)

Given a function \(m_0\in L^\infty(T)\), where \(T\) is the unit disc, the transfer operator (of order \(N\)), or Ruelle operator, \(R\), acts on \(L^1(T)\) by NEWLINE\[NEWLINE (Rf)(z)= 1/N\sum_{w^N=z}|m_0(w)|^2 f(w),\;\;f\in L^1(T), z\in T. NEWLINE\]NEWLINE In multiresolution wavelet analysis the eigenvalues NEWLINE\[NEWLINE h\in L^1(T), \;h\geq 0,\;R(h)=h NEWLINE\]NEWLINE of the operator \(R\) correspond to quadrature wavelet filters and are used to produce the scaling functions. Motivated by this, the formalism of the scaling operator, low pass filter and scaling function is expressed in terms of special representations of the \(C^\ast\)-algebra \(C_N\) on two unitary generators \(U,V\) subject to the relation NEWLINE\[NEWLINE UVU^{-1}=V^{N}. NEWLINE\]NEWLINE It is shown that eigenvalues of the operator \(R\) are in one-to-one correspondence with equivalence classes of some representations of \(C_N\) and the correspondence is explicitly described. These representations may also be viewed as representations of certain (discrete) \(N\)-adic \(ax+b\) groups. This general approach has the advantage of representing the scaling function as the unit function in \(L^2(K_N, \nu)\), where \(K_N\) is certain Bohr compactification of \(R\). Simultaneosly, it allows applications outside wavelet theory. Moreover, results on cocycle equivalence for wavelets filters are given.