@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-NHFK3Q1R-H
  skos:prefLabel "fonction L"@fr, "L-function"@en ;
  a skos:Concept ;
  skos:narrower psr:-Z8FJPBWK-8 .

psr:-RMSGDZWM-G
  skos:prefLabel "fractale"@fr, "fractal"@en ;
  a skos:Concept ;
  skos:narrower psr:-Z8FJPBWK-8 .

psr:-JMNK7BG4-K
  skos:prefLabel "dynamical system"@en, "système dynamique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-Z8FJPBWK-8 .

psr: a skos:ConceptScheme .
psr:-Z8FJPBWK-8
  skos:prefLabel "fonction zêta d'Artin-Mazur"@fr, "Artin-Mazur zeta function"@en ;
  skos:broader psr:-JMNK7BG4-K, psr:-NHFK3Q1R-H, psr:-RMSGDZWM-G ;
  skos:inScheme psr: ;
  skos:definition """In mathematics, the <b>Artin–Mazur zeta function</b>, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals.
<br/>It is defined from a given function <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> as the formal power series
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta _{f}(z)=\\\\exp \\\\left(\\\\sum _{n=1}^{\\\\infty }{\\igl |}\\\\operatorname {Fix} (f^{n}){\\igr |}{\\rac {z^{n}}{n}}\\
ight),}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>ζ<!-- ζ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>f</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>exp</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <munderover>
<br/>              <mo>∑<!-- ∑ --></mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>                <mo>=</mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>              </mrow>
<br/>            </munderover>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow class="MJX-TeXAtom-OPEN">
<br/>                <mo maxsize="1.2em" minsize="1.2em">|</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mi>Fix</mi>
<br/>            <mo>⁡<!-- ⁡ --></mo>
<br/>            <mo stretchy="false">(</mo>
<br/>            <msup>
<br/>              <mi>f</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow class="MJX-TeXAtom-CLOSE">
<br/>                <mo maxsize="1.2em" minsize="1.2em">|</mo>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <msup>
<br/>                  <mi>z</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi>n</mi>
<br/>                  </mrow>
<br/>                </msup>
<br/>                <mi>n</mi>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta _{f}(z)=\\\\exp \\\\left(\\\\sum _{n=1}^{\\\\infty }{\\igl |}\\\\operatorname {Fix} (f^{n}){\\igr |}{\\rac {z^{n}}{n}}\\
ight),}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0aeba71e7f3d982566f388e2acb5b29094f03e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:32.339ex; height:7.509ex;" alt="{\\\\displaystyle \\\\zeta _{f}(z)=\\\\exp \\\\left(\\\\sum _{n=1}^{\\\\infty }{\\igl |}\\\\operatorname {Fix} (f^{n}){\\igr |}{\\rac {z^{n}}{n}}\\
ight),}"></span></dd></dl>
<br/>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\operatorname {Fix} (f^{n})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>Fix</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\operatorname {Fix} (f^{n})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c737b6126168c7d4016ae9f797c7dcaf8fda223" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.74ex; height:2.843ex;" alt="{\\\\displaystyle \\\\operatorname {Fix} (f^{n})}"></span> is the set of fixed points of the <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>th iterate of the function <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle |\\\\operatorname {Fix} (f^{n})|}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo stretchy="false">|</mo>
<br/>        </mrow>
<br/>        <mi>Fix</mi>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msup>
<br/>          <mi>f</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo stretchy="false">|</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle |\\\\operatorname {Fix} (f^{n})|}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf39b4f870753f4de546c1dc02828c248be7885" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.421ex; height:2.843ex;" alt="{\\\\displaystyle |\\\\operatorname {Fix} (f^{n})|}"></span> is the number of fixed points (i.e. the cardinality of that set).
<br/>Note that the zeta function is defined only if the set of fixed points is finite for each <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>. This definition is formal in that the series does not always have a positive radius of convergence.
<br/>The Artin–Mazur zeta function is invariant under topological conjugation.
<br/>The Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is the inverse of the kneading determinant of <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Artin%E2%80%93Mazur_zeta_function">https://en.wikipedia.org/wiki/Artin%E2%80%93Mazur_zeta_function</a>)"""@en ;
  a skos:Concept ;
  dc:modified "2023-08-22"^^xsd:date ;
  dc:created "2023-08-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Artin%E2%80%93Mazur_zeta_function> .