Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2004-01-08
Physica A 340, 219-226 (2004)
Physics
Condensed Matter
Statistical Mechanics
Contribution to the proceedings of 2nd International Conference on News and Expectations in Thermostatistics (NEXT03), Cagliar
Scientific paper
10.1016/j.physa.2004.04.010
We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity $z>1$. First we determine analytically the sensitivity to initial conditions $\xi_{t}$. Then we consider a renormalization group (RG) operation on the partition function $Z$ of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of $Z$ fixes a length-scale transformation factor $2^{-\eta}$ in terms of the generalized dimensions $D_{\beta}$. There exists a gap $\Delta \eta $ in the values of $\eta $ equal to $\lambda _{q}=1/(1-q)=D_{\infty}^{-1}-D_{-\infty}^{-1}$ where $\lambda_{q}$ is the $q$-generalized Lyapunov exponent and $q$ is the nonextensive entropic index. We provide an interpretation for this relationship - previously derived by Lyra and Tsallis - between dynamical and geometrical properties. Key Words: Edge of chaos, multifractal attractor, nonextensivity
Mayoral E.
Robledo Alberto
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