Re: Arnol'd Family


"jmd" <jdroz@xxxxxxx> wrote in message
news:7621767.1129676377006.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
>> The standard Arnol'd map
>>
>> F_A,B= X + A + (B/A)*sin(2*pi*X)
>>
>> Homeo if |B| < or = 1
>> Diffeomorphism |B|<1
>>
>> i want to try and prove those two statements.
>>
>> i was thinking for the Homeomorphism to find the
>> inverse of the function,
>> and show when B is bigger than 1 it is not well
>> defined. but this is proving
>> much more difficult that first thought, and the
>> algebra is getting messy.
>>
>> Which direction should i try and go to prove it for a
>> diffeomorphism??
>> and is there an easier way to prove that is a
>> homeomorphism,
>>
>> Lynn
>>
>>
>
> You are right, finding the inverse explicitly is unpractical (read:
> impossible). However, you can use theorems like the implicit function
> theorem (for the diffeo) and the following (the name of which I can't
> remember): a continuous one-to-one function is open.
>
> You should also check the problem, because according to the definition of
> F you have given, F is not an homeomorphism for B=1/2 and A=1/10!
>
Ah many thanks, using implicit function thm it popped out. but for the
homeo, still working on it. all i need to prove is the end pts when b=+/-
1/2!


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