Entropy computing via integration over fractal measures.

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures. (c) 2000 American Institute of Physics.

[1]  Guarneri,et al.  Multifractal energy spectra and their dynamical implications. , 1994, Physical review letters.

[2]  L. Barreira,et al.  On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. , 1997, Chaos.

[3]  Marius Iosifescu,et al.  Dependence with Complete Connections and its Applications , 1990 .

[4]  H. Schuster.  Deterministic chaos: An introduction , 1984 .

[5]  Giorgio Turchetti,et al.  Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets , 1988 .

[6]  W. D. Withers,et al.  Weight-balanced measures and free energy for one-dimensional dynamics , 1993 .

[7]  Franklin Mendivil,et al.  A classical ergodic property for IFS: a simple proof , 1998 .

[8]  FUNCTIONS OF MARKOV PROCESSES AND ALGEBRAIC MEASURES , 1992 .

[9]  Limit theorems for stochastically perturbed dynamical systems , 1995 .

[10]  Y. Kifer.  Ergodic theory of random transformations , 1986 .

[11]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[12]  David A. Rand,et al.  The entropy function for characteristic exponents , 1987 .

[13]  M. Wolf,et al.  Chaos - The Interplay Between Stochastic and Deterministic Behaviour: Proceedings of the XXXIst Winter School of Theoretical Physics Held in Karpacz, ... February 1995 , 2013 .

[14]  Karol Życzkowski,et al.  Quantum chaos: An entropy approach , 1994 .

[15]  Stable IFSs with probabilities : an ergodic theorem , 1994 .

[16]  L. Olsen,et al.  A Multifractal Formalism , 1995 .

[17]  Örjan Stenflo,et al.  Ergodic Theorems for Iterated Function Systems Controlled by Regenerative Sequences , 1998 .

[18]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[19]  Abraham Boyarsky,et al.  Iterated function systems and dynamical systems. , 1995, Chaos.

[20]  Cohen,et al.  Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems. , 1985, Physical review. A, General physics.

[21]  Abbas Edalat,et al.  Power Domains and Iterated Function Systems , 1996, Inf. Comput..

[22]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

[23]  T. Tél,et al.  Statistical properties of chaos demonstrated in a class of one-dimensional maps. , 1993, Chaos.

[24]  Large-Scale Renormalisation of Fourier Transforms of Self-Similar Measures and Self-Similarity of Riesz Measures , 1996 .

[25]  J. Elton.  An ergodic theorem for iterated maps , 1987, Ergodic Theory and Dynamical Systems.

[26]  Karol Zyczkowski,et al.  Mean Dynamical Entropy of Quantum Maps on the Sphere Diverges in the Semiclassical Limit , 1997, chao-dyn/9707008.

[27]  P. Grassberger,et al.  Estimation of the Kolmogorov entropy from a chaotic signal , 1983 .

[28]  D. Rand.  The singularity spectrum f (α) for cookie-cutters , 1989 .

[29]  Jagat Narain Kapur,et al.  Measures of information and their applications , 1994 .

[30]  H. Crauel,et al.  Iterated Function Systems and Multiplicative Ergodic Theory , 1992 .

[31]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[32]  Karol Zyczkowski,et al.  On the entropy devil's staircase in a family of gap-tent maps , 1998, chao-dyn/9807013.

[33]  Floris Takens,et al.  GENERALIZED ENTROPIES : RENYI AND CORRELATION INTEGRAL APPROACH , 1998 .

[34]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[35]  M. Barnsley,et al.  A new class of markov processes for image encoding , 1988, Advances in Applied Probability.

[36]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[37]  M. Barnsley,et al.  Invariant measures for Markov processes arising from iterated function systems with place-dependent , 1988 .

[38]  L. Olsen,et al.  Random Geometrically Graph Directed Self-Similar Multifractals , 1994 .

[39]  Wojciech Słomczyński,et al.  Coherent states measurement entropy , 1996, chao-dyn/9604010.

[40]  S. Vaienti,et al.  Dynamical integral transform on fractal sets and the computation of entropy , 1993 .

[41]  Generic properties of learning systems , 2000 .

[42]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[43]  J. Yorke,et al.  Correlation dimension for iterated function systems , 1997 .

[44]  Schuster,et al.  Generalized dimensions and entropies from a measured time series. , 1987, Physical review. A, General physics.

[45]  Zyczkowski,et al.  Dynamical entropy for systems with stochastic perturbation , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  A. Lasota.  From fractals to stochastic differential equations , 1995 .

[47]  From quantum entropy to iterated function systems , 1997 .

[48]  K. Falconer.  Techniques in fractal geometry , 1997 .

[49]  K. Życzkowski,et al.  Erratum: Quantum chaos: An entropy approach [J. Math. Phys. 35, 5674–5700 (1994)] , 1995 .