Scaling structure of strange attractors
Physical review. A, General physicsPublished 1 March 1988
Ditza Auerbach, Ben O’Shaughnessy, Itamar Procaccia
Citations29
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
Abstract
The spectrum of scaling exponents of multifractal, hyperbolic strange attractors is quantitatively understood by realizing that each unstable periodic point contributes one scaling exponent to the spectrum. The calculation of the f(\ensuremath{\alpha}) function can thus be reduced to counting periodic orbits.
Keywords
MathematicsPhysics and Astronomy
American Journal of Physics<i>The Fractal Geometry of Nature</i>
21,750 Citations1983Benoît B. Mandelbrot, John Wheeler
Reviews of Modern PhysicsErgodic theory of chaos and strange attractors
4,836 Citations1985Jean‐Pierre Eckmann, David Ruelle
Physical review. A, General physicsFractal measures and their singularities: The characterization of strange sets
3,277 Citations1986Thomas C. Halsey, Mogens H. Jensen +3 more
A description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory, focusing upon the scaling properties of such measures, which are characterized by two indices: \ensuremath{\alpha}, which determines the strength of their singularities; and f, which describes how densely they are distributed.
Physical review. A, General physicsErratum: Fractal measures and their singularities: The characterization of strange sets [Phys. Rev. A<i>33</i>, 1141 (1986)]
2,482 Citations1986Thomas C. Halsey, Mogens H. Jensen +3 more
Physical Review LettersExploring chaotic motion through periodic orbits
515 Citations1987Ditza Auerbach, Predrag Cvitanović +3 more
Physical Review LettersScaling structure of the surface layer of diffusion-limited aggregates
386 Citations1986Thomas C. Halsey, Paul Meakin +1 more
Communications in Mathematical PhysicsThe transition to aperiodic behavior in turbulent systems
324 Citations1980Mitchell J. Feigenbaum
Physical Review LettersGrowth Probability Distribution in Kinetic Aggregation Processes
258 Citations1986C. Amitrano, Antonio Coniglio +1 more
Physical Review LettersTime Ordering and the Thermodynamics of Strange Sets: Theory and Experimental Tests
148 Citations1986Mitchell J. Feigenbaum, Mogens H. Jensen +1 more
From the spectrum of dimensions of a fractal invariant measure of a dynamical system one can extract information about the dynamical process that gave rise to the measure, equivalent to finding the class of Hamiltonians of an Ising model with a given thermodynamics.
Physical review. A, General physicsScaling structure and thermodynamics of strange sets
103 Citations1987Mogens H. Jensen, Leo P. Kadanoff +1 more
A quantitative theory of the scaling properties of Julia sets is presented, using them as a case model for nontrivial fractal sets off the borderline of chaos and showing that generally the theory has a "macroscopic" part which consists of the generalized dimensions of the set, or its spectrum of scaling indexes.
Physical review. A, General physicsScaling of the growth probability measure for fractal structures
40 Citations1986Paul Meakin
