Jeffrey [10] proposed a graphic representation of DNA sequences using Barnsley 's iterative function systems. In spite of further developments in this direction [19, 25, 13], the proposed graphic representation of DNA sequences has been lacking a rigorous connection between its spatial scaling characteristics and the statistical characteristics of the DNA sequences themselves. We 1) generalize Jeffrey's graphic representation to accommodate (possibly infinite) sequences over an arbitrary finite number of symbols, 2) establish a direct correspondence between the statistical characterization of symbolic sequences via R'enyi entropy spectra and the multifractal characteristics (R'enyi generalized dimensions) of the sequences ' spatial representations, 3) show that for general symbolic dynamical systems, the multifractal f H-spectra in the sequence space coincide with the f H-spectra on spatial sequence representations. 1
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450
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Fractals Everywhere
– Barnsley
- 1988
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253
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Fractal Geometry. Mathematical Foundations and Applications
– Falconer
- 1990
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72
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Computation at the onset of chaos
– Crutchfield, Young
- 1990
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67
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The Power of Amnesia
– Ron, Singer, et al.
- 1994
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56
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I.: Mathematical Foundations of Information Theory
– Khinchin
- 1957
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50
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A universal finite memory source
– Weinberger, Rissanen, et al.
- 1995
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37
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Chaos game representation of gene structure
– Jeffrey
- 1990
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28
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Thermodynamics of chaotic systems
– Beck, Schlogl
- 1993
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20
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The study of correlation structures of dna sequences: a critical review
– Li
- 1997
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19
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On the dimension and entropy of probability distributions
– Rényi
- 1959
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18
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Chaos, Dynamics, and Fractals: An Algorithmic Approach to Deterministic Chaos
– McCauley
- 1993
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18
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A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions
– Pesin, Weiss
- 1997
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14
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Affine automata and related techniques for generation of complex images
– Culik, Dube
- 1993
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12
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Entropic profiles of dna sequences through chaos-game-derived images
– Oliver, Bernaola-Galv'an, et al.
- 1993
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11
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On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and lyapunov exponents. multifractal rigidity. Chaos: an Interdisciplinary
– Pesin, Schmeling
- 1997
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9
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Information and Complexity Measures in Dynamical Systems
– GRASSBERGER
- 1991
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8
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Musical structures in data from chaotic attractors
– Mayer-Kress, Bargar, et al.
- 1994
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8
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Entropic feature for sequence pattern through iteration function systems
– Roman-Roldan, Bernaola-Galvan, et al.
- 1994
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8
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Application of information theory to DNA sequence analysis: a review
– Roman-Roldan, Bernaola-Galvan, et al.
- 1996
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8
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Rational and affine expressions for image description
– Culik, Dube
- 1993
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7
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Dimension Theory in Dynamical Systems: Rigorous Results and Applications
– Pesin
- 1997
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6
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Extracting finite state representations from recurrent neural networks trained on chaotic symbolic sequences
– Tino, Koteles
- 1999
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6
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Constructing finite-context sources from fractal representations of symbolic sequences
– Tino, Dorffner
- 1998
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5
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Global fractal dimension of human DNA sequences treated as pseudorandom walks
– Berthelsen, Glazier, et al.
- 1992
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5
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Measures of full dimension on affine invariant sets. Ergodic Theory and Dynamical Systems
– Kenyon, Peres
- 1996
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5
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Quadtrees and the Hausdorff dimension of pictures
– Staiger
- 1989
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3
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Chaos game representation of protein structures
– Fiser, Tusnady, et al.
- 1994
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3
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On the dimension of deterministic and cantor-like sets, symbolic dynamics, and the eckmann-ruelle conjecture
– Pesin, Weiss
- 1996
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3
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Modeling complex sequences with recurrent neural networks
– Tino, Vojtek
- 1998
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2
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Statistical mechanics in biology: how ubiquitous are long-range correlations? Physica A205
– Stanley
- 1996
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2
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A new approach for the classification of functional regions of dna sequences based on fractal representation
– Solovyev, Korolev, et al.
- 1993
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1
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Chaos game representation of proteins
– Basu, Pan, et al.
- 1997
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1
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The evolution of species-type specificity in the global dna sequence organization of mitochondrial genomes
– Hill, Singh
- 1997
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1
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Conditional and realtive multifractal spectra
– Riedi
- 1997
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1
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Additive functions of intervals and hausdorff dimension
– Moran
- 1946
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