are parabolic points in the Mandelbrot set algebraic numbers?
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Define the iterated quadratic polynomial:
$$
beginaligned
f_c^0(z) &= 0 \
f_c^n+1(z) &= f_c^n(z)^2+c
endaligned
$$
The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot set, which has hyperbolic components (cardioid- and disc-like regions) with a rich structure of periodic attractors.
Points $c$ on the boundary of a period $P$ component at the bond point with a period $Pp$ child component satisfy (for some $z$):
$$
beginaligned
f_c^P(z) &= z \
fracpartialpartial zf_c^P(z) &= e^2 pi i fracqp \
gcd(q,p) &= 1
endaligned
$$
Are all algebraic numbers? (I have found some that are.)
Are any algebraic integers? (I have found some that are not.)
What can be said about their degrees (as algebraic numbers) in relation to the $P$ and $p$?
Does the minimal polynomial remain the same when changing (only) $q$ while finding $c$ on the boundary of the same component?
Do points on the boundaries of different components of the same period have different minimal polynomial, even if $p$ and $q$ are the same?
minimal-polynomials complex-dynamics algebraic-numbers
asked Aug 7 at 16:06
Claude
2,488421
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up vote
2
down vote
favorite
Define the iterated quadratic polynomial:
$$
beginaligned
f_c^0(z) &= 0 \
f_c^n+1(z) &= f_c^n(z)^2+c
endaligned
$$
The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot set, which has hyperbolic components (cardioid- and disc-like regions) with a rich structure of periodic attractors.
Points $c$ on the boundary of a period $P$ component at the bond point with a period $Pp$ child component satisfy (for some $z$):
$$
beginaligned
f_c^P(z) &= z \
fracpartialpartial zf_c^P(z) &= e^2 pi i fracqp \
gcd(q,p) &= 1
endaligned
$$
Are all algebraic numbers? (I have found some that are.)
Are any algebraic integers? (I have found some that are not.)
What can be said about their degrees (as algebraic numbers) in relation to the $P$ and $p$?
Does the minimal polynomial remain the same when changing (only) $q$ while finding $c$ on the boundary of the same component?
Do points on the boundaries of different components of the same period have different minimal polynomial, even if $p$ and $q$ are the same?
minimal-polynomials complex-dynamics algebraic-numbers
asked Aug 7 at 16:06
Claude
2,488421
There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Define the iterated quadratic polynomial:
$$
beginaligned
f_c^0(z) &= 0 \
f_c^n+1(z) &= f_c^n(z)^2+c
endaligned
$$
The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot set, which has hyperbolic components (cardioid- and disc-like regions) with a rich structure of periodic attractors.
Points $c$ on the boundary of a period $P$ component at the bond point with a period $Pp$ child component satisfy (for some $z$):
$$
beginaligned
f_c^P(z) &= z \
fracpartialpartial zf_c^P(z) &= e^2 pi i fracqp \
gcd(q,p) &= 1
endaligned
$$
Are all algebraic numbers? (I have found some that are.)
Are any algebraic integers? (I have found some that are not.)
What can be said about their degrees (as algebraic numbers) in relation to the $P$ and $p$?
Does the minimal polynomial remain the same when changing (only) $q$ while finding $c$ on the boundary of the same component?
Do points on the boundaries of different components of the same period have different minimal polynomial, even if $p$ and $q$ are the same?
minimal-polynomials complex-dynamics algebraic-numbers
asked Aug 7 at 16:06
Claude
2,488421
Define the iterated quadratic polynomial:
$$
beginaligned
f_c^0(z) &= 0 \
f_c^n+1(z) &= f_c^n(z)^2+c
endaligned
$$
The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot set, which has hyperbolic components (cardioid- and disc-like regions) with a rich structure of periodic attractors.
Points $c$ on the boundary of a period $P$ component at the bond point with a period $Pp$ child component satisfy (for some $z$):
$$
beginaligned
f_c^P(z) &= z \
fracpartialpartial zf_c^P(z) &= e^2 pi i fracqp \
gcd(q,p) &= 1
endaligned
$$
Are all algebraic numbers? (I have found some that are.)
Are any algebraic integers? (I have found some that are not.)
What can be said about their degrees (as algebraic numbers) in relation to the $P$ and $p$?
Does the minimal polynomial remain the same when changing (only) $q$ while finding $c$ on the boundary of the same component?
Do points on the boundaries of different components of the same period have different minimal polynomial, even if $p$ and $q$ are the same?
minimal-polynomials complex-dynamics algebraic-numbers
asked Aug 7 at 16:06
Claude
2,488421
asked Aug 7 at 16:06
Claude
2,488421
asked Aug 7 at 16:06
Claude
2,488421
asked Aug 7 at 16:06
Claude
2,488421
2,488421
There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39
add a comment |Â
There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39
There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39
There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39
add a comment |Â
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There are also points whose Julia set is a Siegel disk that are also on the boundary of the main cardioid. These have an irrational rotation number for the landing ray. They are not algebraic numbers. They are on the boundary of the m-set. There are uncountable infinite number of them. See math.stackexchange.com/questions/244344/…
– Sheldon L
Aug 8 at 3:39