Why is $x=4$ as a fixed point of a map $sqrt{2}^{x}$ unstable?
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My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x mapsto f(x)$ is given, with $$f(x)= (sqrt{2})^{x}.$$ In that thread, it is stated the map has 2 fixed points, at $x = 2$ and $x=4$. Now to examine their stability we look at the derivative $$f'(x) = (sqrt{2})^{x} log sqrt{2}.$$
Now we have $f'(4) <1$, so $x=4$ should be stable. However, it is claimed, on the contrary, that $x=4$ is, in fact, $textbf{unstable}$. Why is this?
dynamical-systems exponentiation fixed-point-theorems power-towers
asked Nov 7 '18 at 21:37
AlexAlex
71248
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add a comment |
$begingroup$
My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x mapsto f(x)$ is given, with $$f(x)= (sqrt{2})^{x}.$$ In that thread, it is stated the map has 2 fixed points, at $x = 2$ and $x=4$. Now to examine their stability we look at the derivative $$f'(x) = (sqrt{2})^{x} log sqrt{2}.$$
Now we have $f'(4) <1$, so $x=4$ should be stable. However, it is claimed, on the contrary, that $x=4$ is, in fact, $textbf{unstable}$. Why is this?
dynamical-systems exponentiation fixed-point-theorems power-towers
asked Nov 7 '18 at 21:37
AlexAlex
71248
$endgroup$
1
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
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– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26
add a comment |
$begingroup$
My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x mapsto f(x)$ is given, with $$f(x)= (sqrt{2})^{x}.$$ In that thread, it is stated the map has 2 fixed points, at $x = 2$ and $x=4$. Now to examine their stability we look at the derivative $$f'(x) = (sqrt{2})^{x} log sqrt{2}.$$
Now we have $f'(4) <1$, so $x=4$ should be stable. However, it is claimed, on the contrary, that $x=4$ is, in fact, $textbf{unstable}$. Why is this?
dynamical-systems exponentiation fixed-point-theorems power-towers
asked Nov 7 '18 at 21:37
AlexAlex
71248
$endgroup$
My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x mapsto f(x)$ is given, with $$f(x)= (sqrt{2})^{x}.$$ In that thread, it is stated the map has 2 fixed points, at $x = 2$ and $x=4$. Now to examine their stability we look at the derivative $$f'(x) = (sqrt{2})^{x} log sqrt{2}.$$
Now we have $f'(4) <1$, so $x=4$ should be stable. However, it is claimed, on the contrary, that $x=4$ is, in fact, $textbf{unstable}$. Why is this?
dynamical-systems exponentiation fixed-point-theorems power-towers
dynamical-systems exponentiation fixed-point-theorems power-towers
asked Nov 7 '18 at 21:37
AlexAlex
71248
asked Nov 7 '18 at 21:37
AlexAlex
71248
asked Nov 7 '18 at 21:37
AlexAlex
71248
asked Nov 7 '18 at 21:37
AlexAlex
71248
asked Nov 7 '18 at 21:37
AlexAlex
71248
71248
1
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
$endgroup$
– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26
add a comment |
1
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
$endgroup$
– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26
1
1
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
$endgroup$
– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
$endgroup$
– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26
add a comment |
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The question was based on a false premise and is superflous now (see comments below question) as based on a simple error as understood by the OP.
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
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$begingroup$
The question was based on a false premise and is superflous now (see comments below question) as based on a simple error as understood by the OP.
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
$endgroup$
add a comment |
$begingroup$
The question was based on a false premise and is superflous now (see comments below question) as based on a simple error as understood by the OP.
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
$endgroup$
add a comment |
$begingroup$
The question was based on a false premise and is superflous now (see comments below question) as based on a simple error as understood by the OP.
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
$endgroup$
The question was based on a false premise and is superflous now (see comments below question) as based on a simple error as understood by the OP.
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
answered Dec 26 '18 at 10:30
Gottfried HelmsGottfried Helms
23.5k24599
23.5k24599
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1
$begingroup$
Around $x=4$? The derivative is $2 log 2$, or roughly $1.4$.
$endgroup$
– Patrick Stevens
Nov 7 '18 at 21:40
$begingroup$
My apologies! Was doing computations in terms of logs to base 10, rather than natural logarithms. May close the thread. @PatrickStevens
$endgroup$
– Alex
Nov 7 '18 at 21:48
$begingroup$
I'd like to have the question closed/deleted because it's on a false premise. For the case this is not wanted I added a formal answer which is "acceptable" for closing the case.
$endgroup$
– Gottfried Helms
Dec 26 '18 at 10:26