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Takafumi Arakaki  committed 7f9c3dc

thermo-chaos-ja/3262c590feb8

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File thermo-chaos-ja/_sources/dynamical-renyi-entropies.txt

 
 .. [#] 生成的分割とは,無限長の記号列 :math:`i_0, i_1, \cdots` から
        初期値 :math:`x_0` が一意に決まる分割である.
-.. [#] 生成的分割が上限を与えるかについての証明は [Beck1993]_
-       には載っていなかった.
+.. [#] なぜ生成的分割が上限を与えるかについての納得出来る説明は
+       [Beck1993]_ には載っていなかった.
 
        .. todo:: 生成的分割が Rényi エントロピー の上限を与えることに
                  ついての説明を加える

File thermo-chaos-ja/_sources/gibbs-measures-and-srb-measures.txt

 
 .. math::
 
-   J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N)}
+   J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N+1)}
 
 つまり, Gibbs 測度 :math:`\mu_\beta` には
 
    \mu_\beta(J_j^{(N)})
    = \mu_\beta(J_{k_1}^{(N)}) + \cdots
    + \mu_\beta(J_{k_l}^{(N)})
-   = \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N)})
+   = \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N+1)})
 
 の関係式が成立する. 式 :eq:`gibbs-measure-canonical-prob` をこの式に
 代入すれば次の関係式が得られる.
 を得る.
 
 
-.. [#] らしい.
+.. [#] らしい. 力学系の基礎 ([国府2000]_) によれば:
 
-   .. todo:: 「拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する」
-             について,補足する.
+         相空間 :math:`M` 上の力学系 :math:`f: M \to M` がコンパクトな
+         双曲型不変集合 :math:`\Lambda` を持つとき, :math:`\Lambda` が
+         孤立している,すなわち :math:`\Lambda` に孤立近傍が存在するな
+         らば :math:`\Lambda` をマルコフ分割出来ることが知られている.
+         -- p81
+
+       [Robinson1999]_ ([ロビンソン2001]_ の下巻) の VIII.5.1
+       が参考文献になっていた. マルコフ分割自体は,何次元でも
+       条件さえ満たせば可能ってことか.
 
 .. [#] 実は, :math:`\rho(y)` を 自然不変密度 とみなした場合
        (この節の後の方でそうなることを示す), :math:`c_j^{(N)} \to \rho(y)`

File thermo-chaos-ja/_sources/refs.txt

 .. [Bowen2008] Bowen, R. (2008),
    **Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms**.
    Springer.
+.. [国府2000] 国府 寛司. (2000)
+   `力学系の基礎 <http://books.google.com/books?id=RIZDAAAACAAJ>`_.
+.. [Robinson1999] Rex Clark Robinson:
+   `Dynamical systems: stability, symbolic dynamics, and chaos
+   <http://books.google.com/books?id=px5gUpPfww8C>`_, 1999
+.. [ロビンソン2001] C. ロビンソン:
+   `力学系 <http://books.google.com/books?id=6P2W1oV-hgUC>`_, 2001

File thermo-chaos-ja/dynamical-renyi-entropies.html

 <table class="docutils footnote" frame="void" id="id18" rules="none">
 <colgroup><col class="label" /><col /></colgroup>
 <tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id16">[7]</a></td><td><p class="first">生成的分割が上限を与えるかについての証明は <a class="reference internal" href="refs.html#beck1993">[Beck1993]</a>
-には載っていなかった.</p>
+<tr><td class="label"><a class="fn-backref" href="#id16">[7]</a></td><td><p class="first">なぜ生成的分割が上限を与えるかについての納得出来る説明は
+<a class="reference internal" href="refs.html#beck1993">[Beck1993]</a> には載っていなかった.</p>
 <div class="admonition-todo last admonition " id="index-6">
 <p class="first admonition-title">Todo</p>
 <p class="last">生成的分割が Rényi エントロピー の上限を与えることに

File thermo-chaos-ja/gibbs-measures-and-srb-measures.html

 <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー <img class="math" src="_images/math/e66a36b723f7bd5ebc39ab9ff8aa41843855af65.png" alt="J_j^{(N)}"/> の逆像は,
 <img class="math" src="_images/math/f7a1d178284ff1610746d7b8039b834e68949b19.png" alt="(N+1)"/>-シリンダー <img class="math" src="_images/math/723d7b078db6ac4338ca430ec702c48d46c70c1d.png" alt="J_{k_l}^{(N+1)}"/> の和で表せる.</p>
 <div class="math">
-<p><img src="_images/math/78b784199d88c35837103a4601b18196d44ab10d.png" alt="J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N)}" /></p>
+<p><img src="_images/math/173c7183ff2b44610b93a86500a679a92d347193.png" alt="J_j^{(N)} = \bigcup_{\tau=1}^l J_{k_\tau}^{(N+1)}" /></p>
 </div><p>つまり, Gibbs 測度 <img class="math" src="_images/math/e65fb3ce55995c08991faa44d2cfbe6c99762181.png" alt="\mu_\beta"/> には</p>
 <div class="math">
-<p><img src="_images/math/21a80f84f7190d247e558fba39f0960ae890ea21.png" alt="\mu_\beta(J_j^{(N)})
+<p><img src="_images/math/fd83a4eb479c19b80700098b877f3eb8b874b88b.png" alt="\mu_\beta(J_j^{(N)})
 = \mu_\beta(J_{k_1}^{(N)}) + \cdots
 + \mu_\beta(J_{k_l}^{(N)})
-= \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N)})" /></p>
+= \sum_{\tau=1}^l \mu_\beta(J_{k_\tau}^{(N+1)})" /></p>
 </div><p>の関係式が成立する. 式 <a href="#equation-gibbs-measure-canonical-prob">(1)</a> をこの式に
 代入すれば次の関係式が得られる.</p>
 <div class="math">
 </div><p>と書き直すことが出来る.</p>
 <p><img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー <img class="math" src="_images/math/e66a36b723f7bd5ebc39ab9ff8aa41843855af65.png" alt="J_j^{(N)}"/> は <img class="math" src="_images/math/587deaa115238bac53970a4d6e0a68e303834080.png" alt="N \to \infty"/> で
 一点に収束するから, <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダーの集合上の関数 <img class="math" src="_images/math/b27a916783f6e75bec525e8ea688b3d4e7c8cdb1.png" alt="c_j^{(N)}"/>
-は, 相空間上の関数とみなせる. この関数を <img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> とおく <a class="footnote-reference" href="#id6" id="id4">[2]</a> .
+は, 相空間上の関数とみなせる. この関数を <img class="math" src="_images/math/920d76ff898b5f4c18aaae835de2e472405e3133.png" alt="\rho(x)"/> とおく <a class="footnote-reference" href="#id9" id="id4">[2]</a> .
 また, <img class="math" src="_images/math/7de1372f34f371588b83c4c48cf130bc57fb0071.png" alt="\xk[0][j]"/> や <img class="math" src="_images/math/2161d18703853671aad734259e69d4c40e09b928.png" alt="\xk[0][k_\tau]"/> は
 <img class="math" src="_images/math/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" alt="N"/>-シリンダー の収束先で置き換えられる.
 まとめると, <img class="math" src="_images/math/587deaa115238bac53970a4d6e0a68e303834080.png" alt="N \to \infty"/> で</p>
 <table class="docutils footnote" frame="void" id="id5" rules="none">
 <colgroup><col class="label" /><col /></colgroup>
 <tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id3">[1]</a></td><td><p class="first">らしい.</p>
-<div class="admonition-todo last admonition " id="index-2">
-<p class="first admonition-title">Todo</p>
-<p class="last">「拡大的 (expanding) 力学系 ならば,マルコフ分割が存在する」
-について,補足する.</p>
-</div>
+<tr><td class="label"><a class="fn-backref" href="#id3">[1]</a></td><td><p class="first">らしい. 力学系の基礎 (<a class="reference internal" href="refs.html#id2">[国府2000]</a>) によれば:</p>
+<blockquote>
+<div>相空間 <img class="math" src="_images/math/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png" alt="M"/> 上の力学系 <img class="math" src="_images/math/586ea0faa19c8c814fc844a2e69a0bf65f5eabb6.png" alt="f: M \to M"/> がコンパクトな
+双曲型不変集合 <img class="math" src="_images/math/2d295c3b71504a216422e5fffdfbad742c7add64.png" alt="\Lambda"/> を持つとき, <img class="math" src="_images/math/2d295c3b71504a216422e5fffdfbad742c7add64.png" alt="\Lambda"/> が
+孤立している,すなわち <img class="math" src="_images/math/2d295c3b71504a216422e5fffdfbad742c7add64.png" alt="\Lambda"/> に孤立近傍が存在するな
+らば <img class="math" src="_images/math/2d295c3b71504a216422e5fffdfbad742c7add64.png" alt="\Lambda"/> をマルコフ分割出来ることが知られている.
+&#8211; p81</div></blockquote>
+<p class="last"><a class="reference internal" href="refs.html#robinson1999">[Robinson1999]</a> (<a class="reference internal" href="refs.html#id4">[ロビンソン2001]</a> の下巻) の VIII.5.1
+が参考文献になっていた. マルコフ分割自体は,何次元でも
+条件さえ満たせば可能ってことか.</p>
 </td></tr>
 </tbody>
 </table>
-<table class="docutils footnote" frame="void" id="id6" rules="none">
+<table class="docutils footnote" frame="void" id="id9" rules="none">
 <colgroup><col class="label" /><col /></colgroup>
 <tbody valign="top">
 <tr><td class="label"><a class="fn-backref" href="#id4">[2]</a></td><td><p class="first">実は, <img class="math" src="_images/math/f3e17c000d283d99cf4c9d02a982b79d33d0163a.png" alt="\rho(y)"/> を 自然不変密度 とみなした場合

File thermo-chaos-ja/refs.html

 Springer.</td></tr>
 </tbody>
 </table>
+<table class="docutils citation" frame="void" id="id2" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label">[国府2000]</td><td>国府 寛司. (2000)
+<a class="reference external" href="http://books.google.com/books?id=RIZDAAAACAAJ">力学系の基礎</a>.</td></tr>
+</tbody>
+</table>
+<table class="docutils citation" frame="void" id="robinson1999" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label">[Robinson1999]</td><td>Rex Clark Robinson:
+<a class="reference external" href="http://books.google.com/books?id=px5gUpPfww8C">Dynamical systems: stability, symbolic dynamics, and chaos</a>, 1999</td></tr>
+</tbody>
+</table>
+<table class="docutils citation" frame="void" id="id4" rules="none">
+<colgroup><col class="label" /><col /></colgroup>
+<tbody valign="top">
+<tr><td class="label">[ロビンソン2001]</td><td>C. ロビンソン:
+<a class="reference external" href="http://books.google.com/books?id=6P2W1oV-hgUC">力学系</a>, 2001</td></tr>
+</tbody>
+</table>
 </div>
 
 

File thermo-chaos-ja/searchindex.js

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