Ensuring homographic sequences are well-defined
$begingroup$
Given a recursive homographic sequence
$begin{cases} a_0 in mathbb{R} text{ given} \
a_{n+1} = f(a_n) end{cases}$ where $f$ is an homographic function ($x longmapsto dfrac{ax+b}{cx+d}$), how can one determine all values of the first term $a_0$ for which the sequence is well defined.
For specific cases, $a_0$ given, one can just proceed by induction to check that for all $n$, $a_n neq dfrac{-d}{c}$.
My question is whether there is a general result regarding the well-definiteness of homographic sequences.
Thanks.
sequences-and-series recurrence-relations
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
$endgroup$
add a comment |
$begingroup$
Given a recursive homographic sequence
$begin{cases} a_0 in mathbb{R} text{ given} \
a_{n+1} = f(a_n) end{cases}$ where $f$ is an homographic function ($x longmapsto dfrac{ax+b}{cx+d}$), how can one determine all values of the first term $a_0$ for which the sequence is well defined.
For specific cases, $a_0$ given, one can just proceed by induction to check that for all $n$, $a_n neq dfrac{-d}{c}$.
My question is whether there is a general result regarding the well-definiteness of homographic sequences.
Thanks.
sequences-and-series recurrence-relations
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
$endgroup$
add a comment |
$begingroup$
Given a recursive homographic sequence
$begin{cases} a_0 in mathbb{R} text{ given} \
a_{n+1} = f(a_n) end{cases}$ where $f$ is an homographic function ($x longmapsto dfrac{ax+b}{cx+d}$), how can one determine all values of the first term $a_0$ for which the sequence is well defined.
For specific cases, $a_0$ given, one can just proceed by induction to check that for all $n$, $a_n neq dfrac{-d}{c}$.
My question is whether there is a general result regarding the well-definiteness of homographic sequences.
Thanks.
sequences-and-series recurrence-relations
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
$endgroup$
Given a recursive homographic sequence
$begin{cases} a_0 in mathbb{R} text{ given} \
a_{n+1} = f(a_n) end{cases}$ where $f$ is an homographic function ($x longmapsto dfrac{ax+b}{cx+d}$), how can one determine all values of the first term $a_0$ for which the sequence is well defined.
For specific cases, $a_0$ given, one can just proceed by induction to check that for all $n$, $a_n neq dfrac{-d}{c}$.
My question is whether there is a general result regarding the well-definiteness of homographic sequences.
Thanks.
sequences-and-series recurrence-relations
sequences-and-series recurrence-relations
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
edited Dec 10 '18 at 12:21
David G. Stork
10.9k31432
10.9k31432
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
asked Dec 10 '18 at 12:17
Oussama SarihOussama Sarih
47827
47827
add a comment |
add a comment |
1 Answer
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$begingroup$
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence ${y_n}_n$ given by
$$y_{n+1}=f^{-1}(y_n)=frac{dy_n-b}{-cy_n+a} $$
where $y_1=f^{-1}(infty)=-frac dc$
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
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active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence ${y_n}_n$ given by
$$y_{n+1}=f^{-1}(y_n)=frac{dy_n-b}{-cy_n+a} $$
where $y_1=f^{-1}(infty)=-frac dc$
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
$endgroup$
add a comment |
$begingroup$
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence ${y_n}_n$ given by
$$y_{n+1}=f^{-1}(y_n)=frac{dy_n-b}{-cy_n+a} $$
where $y_1=f^{-1}(infty)=-frac dc$
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
$endgroup$
add a comment |
$begingroup$
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence ${y_n}_n$ given by
$$y_{n+1}=f^{-1}(y_n)=frac{dy_n-b}{-cy_n+a} $$
where $y_1=f^{-1}(infty)=-frac dc$
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
$endgroup$
The "forbidden" values of $a_0$ are precisely those occurring in the (possibly finite) sequence ${y_n}_n$ given by
$$y_{n+1}=f^{-1}(y_n)=frac{dy_n-b}{-cy_n+a} $$
where $y_1=f^{-1}(infty)=-frac dc$
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
answered Dec 10 '18 at 19:23
Hagen von EitzenHagen von Eitzen
278k22269501
278k22269501
add a comment |
add a comment |
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