🚀 go-pugleaf

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f_c(x) = x^2 + c$@!"(Jf_c^i(x) = f_c(f_c^(i-1)(x))       if i >= 1 $@$HDj5A$9(J
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$@<B?t$KBP$7$F!"<+A3?t(J n_c(v) $@$r!"(J|f_c^i(0)| >= v $@$H$J$k:G>.$N$b$N$H$7$F(J
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        |      v = 2     |      v = 4     |      v = 8     |      v = 16     
    d   | N=n_c(v)  d*N  | N=n_c(v)  d*N  | N=n_c(v)  d*N  | N=n_c(v)  d*N  
 1.00000|       3 3.00000|       4 4.00000|       5 5.00000|       5 5.00000
 0.10000|      33 3.30000|      34 3.40000|      35 3.50000|      35 3.50000
 0.01000|     315 3.15000|     316 3.16000|     316 3.16000|     317 3.17000
 0.00100|    3143 3.14300|    3144 3.14400|    3145 3.14500|    3145 3.14500
 0.00010|   31417 3.14170|   31418 3.14180|   31419 3.14190|   31419 3.14190
 0.00001|  314160 3.14160|  314161 3.14161|  314162 3.14162|  314162 3.14162

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        (1)  lim  d*n_c(v) = pi
             d->0

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        |      v = 2     |      v = 4     |      v = 8     |      v = 16
    d   |N=n_c(v) sq(d)*N|N=n_c(v) sq(d)*N|N=n_c(v) sq(d)*N|N=n_c(v) sq(d)*N
 1.00000|       2 2.00000|       3 3.00000|       3 3.00000|       4 4.00000
 0.10000|       8 2.52982|       9 2.84605|      10 3.16228|      10 3.16228
 0.01000|      30 3.00000|      31 3.10000|      31 3.10000|      31 3.10000
 0.00100|      97 3.06741|      98 3.09903|      99 3.13065|      99 3.13065
 0.00010|     312 3.12000|     313 3.13000|     314 3.14000|     314 3.14000
 0.00001|     991 3.13382|     992 3.13698|     993 3.14014|     993 3.14014

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	(2)  lim  sqrt(d)*n_c(v) = pi
	     d->0

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  In<4403@balrog.ctron.com> wilson@ctron.com (David Wilson)writes:
>    In this section, I will derive equation (2).  This derivation
>   depends heavily on the fact that the iterates f_c^i(0) are all
>    real.  It is possible that the technique I use could be extended to
>    the complex case, but the generalization is not straightforward.
>
>    In order to derive (2), we need a good estimate of n_c(v) for
>    c = .25+d.  First, we estimate the density of the iterates
>    f_c^i(0) on the real line.
>
>    Let f_c^k(0) = x.  Given that f_c is "well-behaved", which happens
>    to be true for f_c(x) = x^2+.25+d, we would expect the distance
>    between adjacent iterates to vary smoothly, that is, that
>    f_c^(i+1)(0)-f_c^i(0) is near f_c^(k+1)(0)-f_c^k(0) = f(x)-x
>    for i near k.  Since the distance between iterates near x is
>    about f(x)-x, the density of iterates near x is 1/(f(x)-x) per
>    unit.
>
>    This suggests that the number of iterates f_c^k(0) between a and b
>    can be approximated by
>
>	/ b
>	|   1/(f(x)-x) dx
>	/ a
>
>    and that, in particular
>
>                .  / v
>	 n_c(v)  =  |   1/(f(x)-x) dx
>                   / 0
>
>    Remembering that f_c(x) = x^2+.25+d gives
>    
>                .  / v 
>	 n_c(v)  =  |   1/(x^2-x+.25+d) dx
>                   / 0
>
>		= 1/sqrt(d) [ atan((2v-1)/(2sqrt(d)) + atan(1/(2*sqrt(d))) ]
>
>    A bit of calculus yields:
>
>                    .
>	lim  n_c(v) =		4v/(1-2v),	v < 1/2
>	d->0
>
>			    .
>	lim  sqrt(d)*n_c(v) =	pi/2,		v = 1/2
>	d->0
>
>			    .
>	lim  sqrt(d)*n_c(v) =	pi		v > 1/2
>	d->0
>
>    This last equation being the observed phenomenon for v > 2.  We
>    therefore have, for c = .25+d and small d > 0,
>
>               .
>	n_c(v) =	4v/(1-2v),	v < 1/2
>			pi/(2sqrt(v))	v = 1/2
>			pi/sqrt(v)	v > 1/2
>
>    which fairly characterizes n_c(v).
>
>
>David W. Wilson (wilson@ctron.com)

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