Limit behaviour of the box-counting dimension.
$begingroup$
I read this in one of Falconer's books on Fractal Geometry:
The definition of box-counting dimension is essentially saying that
$N_{delta}(F) delta^s to_{delta to 0^{+}} infty$ if $s < dim_B F$ and $N_{delta}(F) delta^s to_{delta to 0^{+}} 0$ if $s > dim_B F$.
However, I'm not sure how to prove this statement.
What I did
I proved that for some $s_0$, the two statements hold.
It remains to prove $s_0 = dim_B F$?
The cases where $dim_B ; F notin {0,infty}$ are clear.
What if $dim_B ; F = 0$ or $dim_B ; F = infty$?
analysis metric-spaces fractals
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
$endgroup$
add a comment |
$begingroup$
I read this in one of Falconer's books on Fractal Geometry:
The definition of box-counting dimension is essentially saying that
$N_{delta}(F) delta^s to_{delta to 0^{+}} infty$ if $s < dim_B F$ and $N_{delta}(F) delta^s to_{delta to 0^{+}} 0$ if $s > dim_B F$.
However, I'm not sure how to prove this statement.
What I did
I proved that for some $s_0$, the two statements hold.
It remains to prove $s_0 = dim_B F$?
The cases where $dim_B ; F notin {0,infty}$ are clear.
What if $dim_B ; F = 0$ or $dim_B ; F = infty$?
analysis metric-spaces fractals
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
$endgroup$
add a comment |
$begingroup$
I read this in one of Falconer's books on Fractal Geometry:
The definition of box-counting dimension is essentially saying that
$N_{delta}(F) delta^s to_{delta to 0^{+}} infty$ if $s < dim_B F$ and $N_{delta}(F) delta^s to_{delta to 0^{+}} 0$ if $s > dim_B F$.
However, I'm not sure how to prove this statement.
What I did
I proved that for some $s_0$, the two statements hold.
It remains to prove $s_0 = dim_B F$?
The cases where $dim_B ; F notin {0,infty}$ are clear.
What if $dim_B ; F = 0$ or $dim_B ; F = infty$?
analysis metric-spaces fractals
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
$endgroup$
I read this in one of Falconer's books on Fractal Geometry:
The definition of box-counting dimension is essentially saying that
$N_{delta}(F) delta^s to_{delta to 0^{+}} infty$ if $s < dim_B F$ and $N_{delta}(F) delta^s to_{delta to 0^{+}} 0$ if $s > dim_B F$.
However, I'm not sure how to prove this statement.
What I did
I proved that for some $s_0$, the two statements hold.
It remains to prove $s_0 = dim_B F$?
The cases where $dim_B ; F notin {0,infty}$ are clear.
What if $dim_B ; F = 0$ or $dim_B ; F = infty$?
analysis metric-spaces fractals
analysis metric-spaces fractals
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
edited Dec 26 '18 at 15:10
Javier
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
asked Dec 26 '18 at 8:52
JavierJavier
2,10521236
2,10521236
add a comment |
add a comment |
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