Robert M. Corless, Gregory W. Frank, and J. Graham Monroe, "Chaos and Continued Fractions", Physica D, 46 1990, pp. 241--253.  Home/Search   Document Not in Database   Summary   Related Articles

....numbers used to represent the iterate. 4 3 Elementary Use However, if we are not so naive, we can use exact or arbitrary precision floatingpoint arithmetic for (some) nonlinear maps. For example, consider exact computation of the first 8192 elements of the Gauss Map starting at x = Gamma 3 [12]. The Gauss map is defined as G(x) ae 0 if x = 0 x Gamma1 mod 1 otherwise and maps [0; 1) to itself. This map is used in the computation of continued fractions; putting n 0 = x] the integer part of x, and fl 0 = x Gamma n 0 , the fractional part of x, and fl k 1 = G(fl k ) then the ....

....the integer part of x, and fl 0 = x Gamma n 0 , the fractional part of x, and fl k 1 = G(fl k ) then the integer parts of 1=fl k are the entries n k in the continued fraction expansion x = n 0 1 n 1 1 n2 1 . If we start with 8500 digits of , then a theorem of Khintchin s [12, 19] says that we may expect that the first 8192 25 partial quotients n k of the computed continued fraction will be correct (this is easy to check afterwards) Then we can give uniformly accurate approximations to the orbit of G starting at Gamma 3 by using the fact that the Gauss map is just the ....

Robert M. Corless, Gregory W. Frank, and J. Graham Monroe, "Chaos and Continued Fractions", Physica D, 46 1990, pp. 241--253.

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