Math 2.0

Dynamical systems and ergodic theory Definition

Ruelle zeta function

Builds on Artin–Mazur zeta functionFixed pointFormal power seriesSubshift of finite type

Definition

Let XX be a set and let f ⁣:X→Xf \colon X \to X be a map. Assume that the set Fix⁡(fn)\operatorname{Fix}(f^n) of fixed points of fnf^n is finite for every n≥1n \ge 1. Let φ ⁣:X→ℂ\varphi \colon X \to \C be a function, called the weight. The Ruelle zeta function of (f,φ)(f, \varphi) is the formal power series ζf,φ(z)=exp⁡(∑n=1∞znn∑x∈Fix⁡(fn)∏k=0n−1φ(fkx)).\zeta_{f, \varphi}(z) = \exp\left( \sum_{n=1}^{\infty} \frac{z^n}{n} \sum_{x \in \operatorname{Fix}(f^n)} \prod_{k=0}^{n-1} \varphi(f^k x) \right).

Remarks

Example: let ff be the shift on the full shift {0,1}ℕ\set{0, 1}^{\N} and let φ=c\varphi = c be a constant. The shift fnf^n has 2n2^n fixed points, each of weight cnc^n, so ζf,φ(z)=exp⁡∑n≥1(2cz)n/n=1/(1−2cz)\zeta_{f, \varphi}(z) = \exp \sum_{n \ge 1} (2cz)^n / n = 1/(1 - 2cz).

Non-example: for the identity map of the circle the set Fix⁡(f)\operatorname{Fix}(f) is infinite, so the inner sum is not defined.

With φ=1\varphi = 1 one gets the Artin–Mazur zeta function of ff. More generally the weight may take values in d×dd \times d matrices, and then the product is replaced by the trace of the ordered product φ(fn−1x)⋯φ(x)\varphi(f^{n-1} x) \cdots \varphi(x). If ff is the shift on a subshift of finite type with transition matrix AA and φ(x)=w(x0,x1)\varphi(x) = w(x_0, x_1) depends only on the first two symbols, then ζf,φ(z)=1/det⁡(I−zB)\zeta_{f, \varphi}(z) = 1/\det(I - z B), where Bij=Aijw(i,j)B_{ij} = A_{ij} w(i, j).

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  1. Artin–Mazur zeta function concept
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  4. Subshift of finite type concept
  5. Ruelle zeta function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ruelle_zeta_function
About this page
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WritingW0 stub
MSC37C30, 11M41
DefinesRuelle zeta function
UsesArtin–Mazur zeta function, Fixed point, Formal power series, Subshift of finite type

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