Homework 11

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There is no programming assignment this week.

Q11.1 \(\quad\) "Golden mean" pruned map (Exercise 22.1)

We have tried to study the symbolic dynamics of "Golden mean" pruned map (figure below) before and we conclude that symbolic sequence \(\_00\_\) is pruned. To simplify the discussion, we define a new symbol \(2 = 01\), then the complete alphabet for the symbolic dynamics in this system is \(\{1, 2\}\). Any allowed itinerary is an unrestricted combination of these two symbols. Therefore, the trace formula (21.7) reduces to \[ tr \mathcal{L}^n = \sum_{m=0}^{n} \binom{n}{m} {1 \over \left\vert 1-\Lambda_0^{m} \Lambda_1^{n-m} \right\vert } = \sum_{k=0}^{\infty} \left( {1 \over |\Lambda_0| \Lambda_0^{k}} + {1 \over |\Lambda_1| \Lambda_1^{k}} \right)^n \,. \] Here \(\Lambda_0 = -\Lambda\) and \(\Lambda_1 = -\Lambda^2\) come from the contributions of orbit parts with symbols \(1\) and \(2\) to the trace respectively ( You should know why ). Note, \(\Lambda=(\sqrt{5}+1)/2\). This is very similar to the piecewise-linear map. Please consult Example 21.2 to understand the above expression and confirm that the identity is mathematically correct.

Now turn to formula (22.3), which reveals the relation between trace and spectral determinant of the evolution operator. Try to obtain the explicit form of spectral determinant. What is the coefficient of term \(z\) in the spectral determinant ? Note the expression for trace given above is for the "new" system \( \{1, 2\}\). Here, we are interested in the dynamics in the "original" golden map. The only differenece is the topological length for symbol 2, which is 2 in the "orgianal" system, so \(z^2\) should be assign to the part of orbit which has symbol 2.

Q11.2 \(\quad\) dynamic zeta function (Exercise 22.2)

For the piecewise-linear map, the dynamical zeta function is \[ 1/\zeta(z) = \prod_{p} \left( 1 - \frac{z^{n_{p}}}{|\Lambda_p|} \right) \,. \] Here, the product includes all prime periodic orbits. For this system, we can get a closed form of dynamical zeta function. I recommend two approaches.

First, you can enumerate all short periodic orbits inside this system and substitute them into the above formula, remove terms that cancel out each other, then you are left with the dynamical zeta function.

Second, formula (22.28) reveals the relation between zeta function and spectral determinant in this system. Also the spectral determinant is given in example 22.1. therefore, some simple mathematical derivation can produce the dynamical zeta function.

After obtaining the dynamical zeta function, try to get its leading eigenvalue. What is it ? Your result should make sense, because the escape rate in this system is zero. ( Note, please input a decimal number )

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golden map