A098081 - OEIS

%I #5 Mar 30 2012 17:34:14

%S 0,7,1,18,0,55,1,295,0,2412,1,28397,0,455264,1,9487328,0,247621547

%N a[n]=1+1/Log[Gamma[2-a[n-1]]]:

%C My idea is to invent a fractional Lyapunov exponent: L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m where: Gamma[2]=1 d^sx[n]/dt^s=Gamma[2]*x[n]^(1-s)/Gamma[2-s] Or L(s)=Sum[ Log[Abs[x[n]^(1-s)/Gamma[2-s]],{n,1,m}]/m If the average of x[n] is one then or Log[1]=0: L[s]=Log[Gamma[2-s]] which is always negative since: 0<=Gamma[2-s]<=1 on 0<=s<=1 which gives a Kaplan -Yorke dimension of: (d0 the topological dimension) if the other exponents are one dky(s)=d0+1/Log[Gamma[2-s]] Which gives a roughness gap in the range: s'=1+1/Log[Gamma[2-s]]

%D Kiran M. Kolwankar and Anil D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6 (1996), 505-513.

%t Clear[s, n, digits] s[n_]:=1+1/Log[Gamma[2-s[n-1]]]; s[1]=0.6 digits=18 a=Table[Floor[Abs[s[n]]], {n, 1, digits}] ListPlot[a, PlotJoined->True, PlotRange->All, Axes-> False]

%K nonn

%O 0,2

%A _Roger L. Bagula_, Sep 13 2004