Fixed points of the Ruelle-Thurston operator and the Cauchy transform

The Milnor-Thurston determinant and the Ruelle transfer operator

The topological entropy $h_{\rm top}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that $\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious …

扩张 Thurston 映射的素轨道定理：Lattès 映射与分裂 Ruelle 算子

Prime orbit theorems for expanding Thurston maps: Lattès maps and split Ruelle operators

本文为二维球面上的扩张 Thurston 映射建立了素数定理的类比，证明在满足 α-强不可积性条件的权重下，原始周期轨道数渐近服从对数积分并带指数误差界，结果覆盖后临界有限有理映射，并对 Lattès 映射给出更强结论。

On the distribution of zeros of a Ruelle zeta-function

We study the limit distribution of eigenvalues of a Ruelle operator (which is also called the Thurston pushforward operator) for the dynamical system $z \mapsto z^2+c$ when $c<-2$ and tends to $-2$.

Dynamical zeta functions

These are notes from a course given in Orsay in 2002 explaining carefully the Milnor-Thurston kneading determinant approach to dynamical zeta functions as interpreted by Baladi and Ruelle (Invent. Math. 1996). We make them available in view of the recent renewed interest in this approach. (see arXiv:1501.00294, The Milnor-Thurston determinant and the Ruelle transfer operator, HH Rugh, Comm. Math. Phys. 342 (2016) 603-614, and arXiv:1407.5313, Kneading with weights, HH Rugh, Lei Tan, J. Fractal Geom. 2 (2015) 339-375)