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• W3204257416 abstract "We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f colon left-parenthesis upper X comma x 0 right-parenthesis right-arrow left-parenthesis upper X comma x 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>fcolon (X,x_0)to (X,x_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complex surface having <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 0> <mml:semantics> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>x_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a normal singularity. We prove that as long as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 0> <mml:semantics> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>x_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not a cusp singularity of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then it is possible to find arbitrarily high modifications <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi colon upper X Subscript pi Baseline right-arrow left-parenthesis upper X comma x 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>π<!-- π --></mml:mi> </mml:msub> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>pi colon X_pi to (X,x_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the dynamics of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (or more precisely of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript upper N> <mml:semantics> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>f^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> big enough) on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Subscript pi> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>π<!-- π --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>X_pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is algebraically stable. This result is proved by understanding the dynamics induced by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a space of valuations associated to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in fact, we are able to give a strong classification of all the possible dynamical behaviors of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on this valuation space. We also deduce a precise description of the behavior of the sequence of attraction rates for the iterates of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we prove that in this setting the first dynamical degree is always a quadratic integer." @default.
• W3204257416 created "2021-10-11" @default.
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• W3204257416 date "2021-07-01" @default.
• W3204257416 modified "2023-09-24" @default.
• W3204257416 title "Local dynamics of non-invertible maps near normal surface singularities" @default.
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• W3204257416 doi "https://doi.org/10.1090/memo/1337" @default.
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