https://tetrationforum.org/hyperops_wiki/index.php?action=history&feed=atom&title=Super-attracting_fixpoint 2026-08-10T18:00:12Z Revision history for this page on the wiki MediaWiki 1.35.8 https://tetrationforum.org/hyperops_wiki/index.php?title=Super-attracting_fixpoint&diff=109&oldid=prev 2011-06-07T12:47:59Z

<p>uniqueness</p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:47, 7 June 2011</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A fixpoint $a$ of $f$ such that $f&#039;(a)=0$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A fixpoint $a$ of $f$ such that $f&#039;(a)=0$.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the [[Böttcher equation]] $\beta(f(z))=\beta(<del class="diffchange diffchange-inline">x</del>)^p$ is satisfied. $\beta$ is called the [[Böttcher coordinate]].</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the [[Böttcher equation]]  </div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">$</ins>$\beta(f(z))=\beta(<ins class="diffchange diffchange-inline">z</ins>)^p<ins class="diffchange diffchange-inline">,$$</ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">with $\beta(0)=0$, $\beta&#039;(0)\neq 0</ins>$ is satisfied. $\beta$ <ins class="diffchange diffchange-inline">is unique up to taking positive integer powers and multiplication by ($p-1$)-root of unity. It </ins>is called the [[Böttcher coordinate]].</div></td></tr> </table>

Bo198214 https://tetrationforum.org/hyperops_wiki/index.php?title=Super-attracting_fixpoint&diff=106&oldid=prev 2011-06-07T12:11:15Z

<p>initial draft</p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:11, 7 June 2011</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A fixpoint $<del class="diffchange diffchange-inline">p</del>$ of $f$ such that $f&#039;(<del class="diffchange diffchange-inline">p</del>)=0$.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A fixpoint $<ins class="diffchange diffchange-inline">a</ins>$ of $f$ such that $f&#039;(<ins class="diffchange diffchange-inline">a</ins>)=0$.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">See also </del>[[<del class="diffchange diffchange-inline">Boettcher Equation</del>]]<del class="diffchange diffchange-inline">, </del>[[<del class="diffchange diffchange-inline">Boettcher coordinates</del>]].</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the </ins>[[<ins class="diffchange diffchange-inline">Böttcher equation</ins>]] <ins class="diffchange diffchange-inline">$\beta(f(z))=\beta(x)^p$ is satisfied. $\beta$ is called the </ins>[[<ins class="diffchange diffchange-inline">Böttcher coordinate</ins>]].</div></td></tr> </table>

Bo198214 https://tetrationforum.org/hyperops_wiki/index.php?title=Super-attracting_fixpoint&diff=84&oldid=prev 2011-06-05T12:04:47Z

<p>draft</p> <p><b>New page</b></p><div>A fixpoint $p$ of $f$ such that $f&#039;(p)=0$.<br /> See also [[Boettcher Equation]], [[Boettcher coordinates]].</div>

Bo198214