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<title>Wandernder Punkt</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Wandernder Punkt</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der Theorie der <a href="Dynamisches_System" title="Dynamisches System">dynamischen Systeme</a> ist ein <b>nicht wandernder Punkt</b> (auch: <b>nichtwandernder Punkt</b>) ein Punkt, dessen <a href="Orbit_(Mathematik)" class="mw-redirect" title="Orbit (Mathematik)">Orbit</a> wieder beliebig nahe an die Ausgangsposition zurückkehrt und auch die Orbiten einer ganzen Umgebung des Punktes wieder beliebig nahe an diesen Punkt herankommen. (Insbesondere sind <a href="Periodischer_Orbit" title="Periodischer Orbit">periodische Punkte</a> nichtwandernd.) Entsprechend ist ein <b>wandernder Punkt</b> ein Punkt, für den eine ganze Umgebung nie wieder in diese Umgebung zurückkehrt. Die Menge der wandernden bzw. nicht wandernden Punkte wird als <b>wandernde Menge</b> bzw. <b>nichtwandernde Menge</b> bezeichnet.

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<div class="mw-heading mw-heading2"><h2 id="Definition_für_diskrete_dynamische_Systeme"><span id="Definition_f.C3.BCr_diskrete_dynamische_Systeme"></span>Definition für diskrete dynamische Systeme</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein <a href="Metrischer_Raum" title="Metrischer Raum">metrischer Raum</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to X}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/773c98269d8a3c82725e5cb650243666484b528b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.887ex; height:2.509ex;" alt="{\displaystyle f\colon X\to X}" loading="lazy"></span> eine <a href="Stetige_Abbildung" class="mw-redirect" title="Stetige Abbildung">stetige Transformation</a>.

</p><p>Ein Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span> ist ein wandernder Punkt, wenn es eine <a href="Umgebung_(Mathematik)" title="Umgebung (Mathematik)">Umgebung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy"></span> gibt, so dass

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle f^{n}(U)\cap U=\emptyset }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbb69cabf1d6544660fc232fd4448f44de2770c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.757ex; height:2.843ex;" alt="{\displaystyle f^{n}(U)\cap U=\emptyset }" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"></span>.

</p><p>Ein Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"></span> ist ein nichtwandernder Punkt, wenn es für jede Umgebung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy"></span> ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"></span> mit

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f^{n}(U)\cap U\not =\emptyset }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/322246f1709ea848f51ec7fbc3dd9f137729e5ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.757ex; height:2.843ex;" alt="{\displaystyle f^{n}(U)\cap U\not =\emptyset }" loading="lazy"></span></dd></dl>

<p>gibt.

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<div class="mw-heading mw-heading2"><h2 id="Definition_für_Flüsse"><span id="Definition_f.C3.BCr_Fl.C3.BCsse"></span>Definition für Flüsse</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eine <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeit</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon M\times \mathbb {R} \to M}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi \colon M\times \mathbb {R} \to M}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e285ac9a18472f728d4293d8efe3aee6fa51a6ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.436ex; height:2.509ex;" alt="{\displaystyle \phi \colon M\times \mathbb {R} \to M}" loading="lazy"></span> ein <a href="Fluss_(Mathematik)" title="Fluss (Mathematik)">Fluss</a>.

</p><p>Ein Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span> ist ein wandernder Punkt, wenn es eine <a href="Umgebung_(Mathematik)" title="Umgebung (Mathematik)">Umgebung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> und ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T>0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e39e90b9e1e7d5be3b5eae57729dc63494bbe3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T>0}" loading="lazy"></span> gibt, so dass

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<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{t}(U)\cap U=\emptyset }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8db82000f4ba74787665dcd526db52d2cb0bdf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.429ex; height:2.843ex;" alt="{\displaystyle \phi _{t}(U)\cap U=\emptyset }" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle t\geq T}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35612e99f3705e8109c493a598021d88a2093f42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.574ex; height:2.343ex;" alt="{\displaystyle t\geq T}" loading="lazy"></span>.

</p><p>Ein Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span> ist ein nichtwandernder Punkt, wenn es für jede Umgebung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy"></span> und für jedes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T>0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e39e90b9e1e7d5be3b5eae57729dc63494bbe3fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T>0}" loading="lazy"></span> ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\geq T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\geq T}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35612e99f3705e8109c493a598021d88a2093f42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.574ex; height:2.343ex;" alt="{\displaystyle t\geq T}" loading="lazy"></span> mit

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<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \phi _{t}(U)\cap U\not =\emptyset }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/751f0ade5fd1b9ac76bbde07010258889cec384b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.429ex; height:2.843ex;" alt="{\displaystyle \phi _{t}(U)\cap U\not =\emptyset }" loading="lazy"></span></dd></dl>

<p>gibt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<p>Die Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (f)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Omega (f)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d15d1fbe78c21d34f1d9e3ecd64b09fc275ef971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.766ex; height:2.843ex;" alt="{\displaystyle \Omega (f)}" loading="lazy"></span> der nichtwandernden Punkt ist <a href="Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a>, <a href="Invariante_Menge" title="Invariante Menge">invariant</a> und enthält alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>-<a href="Limesmenge" title="Limesmenge">Limesmengen</a>. Sie enthält alle <a href="Rekurrenter_Punkt" title="Rekurrenter Punkt">rekurrenten Punkte</a>, es muss aber nicht jeder nichtwandernde Punkt auch rekurrent sein.

</p><p>Wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> <a href="Kompakter_Raum" title="Kompakter Raum">kompakt</a> ist, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (f)\not =\emptyset }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega (f)\not =\emptyset }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de4521af2a9cda2ea9c63774cae3b7d669ce2d2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.027ex; height:2.843ex;" alt="{\displaystyle \Omega (f)\not =\emptyset }" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Manfred_Denker" title="Manfred Denker">Manfred Denker</a>: <i>Einführung in die Analysis dynamischer Systeme.</i> Springer-Lehrbuch. Springer-Verlag, Berlin 2005, ISBN 3-540-20713-9.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Nonwandering.html">Nonwandering</a> (MathWorld)</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2021-07-08" href="https://de.wikipedia.org/wiki/?title=Wandernder_Punkt&oldid=213667965">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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