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  <div class="section" id="perron-frobenius">
<h1>Perron-Frobenius 演算子<a class="headerlink" href="#perron-frobenius" title="Permalink to this headline"></a></h1>
<div class="admonition-todo admonition " id="index-0">
<p class="first admonition-title">Todo</p>
<p class="last">Perron-Frobenius 演算子の導出</p>
</div>
<p id="index-1"><strong>Perron-Frobenius 演算子</strong> (<strong>Perron-Frobenius operator</strong>)
<img class="math" src="_images/math/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png" alt="L"/> の定義:</p>
<div class="math">
<p><img src="_images/math/aa3aebb3fd5eeaf559c76fd0a12a4ec8699059f0.png" alt="L \rho(y) := \sum_{x \in f^{-1}(y)} \frac{\rho(x)}{|f'(x)|}" /></p>
</div><p id="index-2"><strong>一般化 Perron-Frobenius 演算子</strong> (<strong>generalized Perron-Frobenius operator</strong>)
<img class="math" src="_images/math/de436cc6099e42eb0acf3e3bfb244509e11bd0c7.png" alt="L_\beta"/> の定義:</p>
<div class="math">
<p><img src="_images/math/09112be8151c901d3380047649ea0b282e0de294.png" alt="L_\beta \, \rho(y) := \sum_{x \in f^{-1}(y)} \rho(x) |f'(x)|^{-\beta}" /></p>
</div><p>Perron-Frobenius 演算子 の固有値 <img class="math" src="_images/math/51dad21033dead1c49732748e32c13252ae0b25a.png" alt="\eta_\alpha"/>固有関数 <img class="math" src="_images/math/76c3f7c80b0892c38cf08afb27bff982fe62b028.png" alt="\phi_\alpha (x)"/> は以下の関係を満たす:</p>
<div class="math">
<p><img src="_images/math/3c3705dba9774b1de8b100ebea3f162ffa6c5690.png" alt="L_\beta \, \phi_\alpha(x) = \eta_\alpha \phi_\alpha(x)" /></p>
</div><p>自然不変密度は Perron-Frobenius 演算子の最大固有値 <img class="math" src="_images/math/c47989c62702ef22a59a8bcd18e5a1290b19ec54.png" alt="\eta_0 = 1"/>
に対応する固有関数 <img class="math" src="_images/math/254c8ed9835a01428975acb16206391263c66bdf.png" alt="\phi_0 (x)"/> である.</p>
<div class="admonition-todo admonition " id="index-3">
<p class="first admonition-title">Todo</p>
<p class="last">なぜ (あるいはどのような条件で) Perron-Frobenius 演算子の
最大固有値 <img class="math" src="_images/math/67a66635f0821bb52d4d2301841f796ccf8bb2ad.png" alt="\eta_0"/> は 1 なのか?
固有関数 <img class="math" src="_images/math/254c8ed9835a01428975acb16206391263c66bdf.png" alt="\phi_0 (x)"/> が重解 (縮退している) 場合は無いのか?
あるとしたらどういう場合か?
などについてまとめる.</p>
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