revans@euclid.ucsd.edu (Ron Evans) (03/29/91)
Consider the sequence x , x ,... defined by x = 0,
0 1 0
2
x = x - 3/2 . TRUE OR FALSE: lim inf x = - 3/2 ?
n+1 n n
To provide some incentive to solve this, let me say, yes,
this is a homework problem.
Perhaps the best chance for a solution lies with a powerful
Russian mathematician, so I hope this posting reaches Harvard.
Ron Evans (revans@math.ucsd.edu)
Ron Evans, Department of Mathematics, UCSD (revans@math.ucsd.edu)scavo@cie.uoregon.edu (Tom Scavo) (03/30/91)
In article <5071@network.ucsd.edu> revans@euclid.UUCP (Ron Evans) writes: > > >Consider the sequence x , x ,... defined by x = 0, > 0 1 0 > 2 >x = x - 3/2 . TRUE OR FALSE: lim inf x = - 3/2 ? > n+1 n n > In article <1991Mar26.215525.27585@ariel.unm.edu> I gave a reference and wrote: > >(iii) For -3/4 < c < 1/4 , Q has a pair of fixed points x1 and x2 >given by > > 1 + sqrt(1 - 4c) 1 - sqrt(1 - 4c) > x1 = ---------------- and x2 = ---------------- > 2 2 > >with x1 repelling and x2 attracting; the basin of attraction for x2 >is (-x1, x1). For c outside this range (except perhaps at the endpoints), Q (where Q(x) = x^2 + c ) has no attracting fixed points. So, for c = -3/2 the answer is going to be FALSE no matter what value you write down! A more interesting question would be: is the orbit of 0 periodic when c = -3/2 ? In article <1991Mar29.141108.8914@athena.cs.uga.edu> is@athena.cs.uga.edu (Bob Stearns) writes: >I am neither Russian nor a powerful mathematician, but a simple test with >my local computer indicates that the iterated function x = x^2-3/2 is >periodic with period 254 or less. I'm not sure what you mean by this. Is it periodic, and if so, what is the period? Does anyone know? In article <5071@network.ucsd.edu> revans@euclid.UUCP (Ron Evans) continues: > Perhaps the best chance for a solution lies with a powerful >Russian mathematician, so I hope this posting reaches Harvard. You're probably referring to Sarkovskii's theorem, but I don't see what that has to do with the original question? And what does Harvard have to do with anything? Try not to be so vague next time. Tom Scavo scavo@cie.uoregon.edu
scavo@cie.uoregon.edu (Tom Scavo) (03/30/91)
The article I posted previously is pure garbage. That's what
I get for not reading the original article more closely. The
lim inf is indeed finite as anyone can see just by looking at
^^^
the orbit diagram for the map Q(x) = x^2 + c . But I don't
know how to compute it analytically.
Tom Scavo
scavo@cie.uoregon.edu