Fractal Geometry: Mathematical Foundations and Applications.

1990/09/01 by K. Falconer, K. J. Falconer · 6,029 citations
Mathematics · #Brownian motion #Computer science #Dimension function #Dynamical systems theory #Effective dimension #Fractal #Fractal analysis #Fractal dimension #Fractal dimension on networks #Fractal landscape #Fractional Brownian motion #Hausdorff dimension #Hausdorff measure #Julia set #Mandelbrot set #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Measure (data warehouse) #Multifractal system

paper · doi:10.2307/2532125

published in Biometrics 46(3), 886 (Oxford University Press)

openalex publication_date 1990/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applica

Cited by

• The Hausdorff dimension of the projections of self-affine carpets
• Spectra of a class of Cantor–Moran measures with three-element digit sets
• Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices
• Metric properties of the product of consecutive partial quotients in continued fractions
• Quantifying preferential flow in soils: A review of different techniques
• Neural ODE Control for Classification, Approximation, and Transport
• Long-time behavior of second order evolution equations with nonlinear damping
• Minkowski dimension and content of complex continued fractions
• Descriptions of Cantor Sets: A Set‑Theoretic Survey and Open Problems
• Bad approximability, bounded ratios and Diophantine exponents
• Fractal Turán-Nazarov Inequality and Observability for Schrödinger Equations
• Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents
• Fractal analysis of hyperbolic and nonhyperbolic fixed points and singularities of dynamical systems in ℝⁿ
• Badly approximable numbers and Littlewood-type problems
• The Dimensions of Individual Strings and Sequences
• Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps
• Rethinking Intrinsic Dimension Estimation in Neural Representations
• Box-Counting Dimension Using Triangles
• Fractal measures of image local features: an application to texture recognition
• Boundary behavior of solutions of a class of genuinely nonlinear hyperbolic systems
• A Fibonacci control system with application to hyper-redundant manipulators
• Bifurcation sets arising from non-integer base expansions
• The Geometric Invariants of Null Cartan Curves Under The Similarity Transformations
• Curves That Must Be Retraced
• Invariant Measures with Bounded Variation Densities for Piecewise Area Preserving Maps
• From Pen to Palette: Mathematical Analysis of van Gogh's Psychological Trajectory
• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
• C^*-algebras associated with self-similar sets
• Dimension results for inhomogeneous Moran set constructions
• Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential
• Algorithms to test open set condition for self-similar set related to P.V. numbers
• Schmidt Games and Nondense forward Orbits of certain Partially Hyperbolic Systems
• Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation
• Latent Stability Analysis of Malware Representations Under Feature-Space Perturbations
• ROSE: A Retinal OCT-Angiography Vessel Segmentation Dataset and New Model
• Fractal dimensions of graph of Weierstrass-type function and local Hölder exponent spectra
• The fractal geometry of opinion formation
• Quantization Dimension of 1-variable Random Self-Similar Measures
• Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes
• Dimension estimates for C¹ iterated function systems and repellers. Part II
• Self-similar sets, simple augmented trees, and their Lipschitz equivalence
• Hausdorff dimension in R-analytic profinite groups
• Processes iterated ad libitum
• Anisotropic covering of fractal sets
• A winning approach to the intersections of twisted non-recurrent sets with fractals
• Rational points in Cantor sets in the complex plane
• Infinite Dimensional Multifractal Analysis of the Wiener measure
• Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets
• On the bifurcation set of unique expansions
• Restricted mean value theorems and metric theory of restricted Weyl sums
• Strong-field gravitational lensing by black holes
• Multifractal Recalibration of Neural Networks for Medical Imaging Segmentation
• Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions
• Variational principle for weighted topological pressure
• Spectrality and non-spectrality of a class of Moran measures with three-element digits
• Fractal Analysis of Pseudo Foci in Piecewise Analytic Systems
• Entropy-scaling search of massive biological data
• Periodic modulation of the space-filling nature of the turbulent flame leads to spiky heat release oscillations
• Twisted Diophantine approximation for matrix transformations of tori
• Hausdorff Measure and Dimension with Examples
• Dichotomy for the Hausdorff dimension of the set of nonergodic directions
• Remez-Type Inequality for Discrete Sets
• On optimal cover and its possible shape for fractals embedded into 2D Euclidian space
• Two-Scale Frostman Measures
• Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension
• Discrete Scale Invariance in the Cascade Heart Rate Variability Of Healthy Humans
• The grounding for Continuum
• Real Projective Iterated Function Systems
• On metric, dynamical, probabilistic and fractal phenomena connected with the second Ostrogradsky expansion
• Combinatorial topology of three-dimensional self-affine tiles
• The Hausdorff dimension of the range of the Lévy multistable processes
• Experimental observation of fractal modes in unstable optical resonators
• Twofold Cantor sets in R
• Local dimension for piecewise monotonic maps on the interval
• Finite-dimensional global attractors in Banach spaces
• Boundary multifractal behaviour for harmonic functions in the ball
• Minkowski and packing Dimension comparisons for sets with Reifenberg properties
• Random fractals and tree-indexed Markov chains
• Fractal and Smoothness Properties of Space-Time Gaussian Models
• Variational principles for topological entropies of subsets
• Pursuit Fractal Analysis of Time-Series Data
• Fractality of Massive Graphs: Scalable Analysis with Sketch-Based Box-Covering Algorithm
• Erdős Semi-groups, arithmetic progressions and Szemerédi's theorem
• Rational points near self-similar sets
• Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review
• Chaotic and Topological Properties of Continued Fractions
• On Khintchine exponents and Lyapunov exponents of continued fractions
• On a problem in simultaneous Diophantine approximation: Schmidt's conjecture
• Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension
• A set-oriented path following method for the approximation of parameter dependent attractors
• Calculus on fractal subsets of real line - I: formulation
• The topological property of the irregular sets on the lengths of basic intervals in beta-expansions
• An algorithm for the word entropy
• Diophantine approximations on random fractals
• The dead leaves model : general results and limits at small scales
• Martin boundary theory on inhomogenous fractals
• Analysis of Fractal Dimension of Mixed Riemann-Liouville Fractional Integral
• Contour Models for Boundaries Enclosing Star-Shaped and Approximately Star-Shaped Polygons
• Knapp-type obstructions in multilinear fractal Fourier extension
• On Convexity, Mid-Point Convexity and Hausdorff Measures of Sets
• An L⁴ maximal estimate for quadratic Weyl sums
• The range of dimensions of microsets
• Codimension formulae for the intersection of fractal subsets of Cantor spaces
• Some Fundamental Theorems in Mathematics
• The Hentschel-Procaccia Conjecture
• A fractal perspective on optimal antichains and intersecting subsets of the unit n-cube
• Notes on metrics, measures, and dimensions
• Larger Hausdorff Dimension in Scanning Pattern Facilitates Mamba-Based Methods in Low-Light Image Enhancement
• The Problem of Small Unilateral Deviations: the Existence of Decay Exponents
• Parameter Families of Iterated Function Systems and Continuity
• THE POINTWISE DENSITIES OF NON-SYMMETRIC CANTOR SETS
• Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence
• Exact Solutions for Classes of Nonlinear Differential Equations on Fractal Supports
• T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning
• On arithmetic sums of fractal sets in \Bbb R^d
• Deep Unfolded Recovery of Sub-Nyquist Sampled Ultrasound Image
• Correlation Dimension of Auto-Regressive Large Language Models
• Volume growth of quasihyperbolic balls
• On the gap between cluster dimensions of loop soups on ℝ³ and the metric graph of ℤ³
• An explanatory model to validate the way water activity rules periodic terrace generation in Proteus mirabilis swarm
• Nonlocal p-Laplacian Variational problems on graphs
• Large-scale stochastic propagation method beyond the sequential approach
• On Bernoulli convolutions generated by second Ostrogradsky series and their fine fractal properties
• New dimension bounds for αβ sets
• Non-smooth atomic decompositions, traces on Lipschitz domains, and\n pointwise multipliers in function spaces
• Hausdorff dimensions of level sets related to moving digit means
• A new regularization possibility for the Boltzmann equation with soft potentials
• Heaviness in Circle Rotations
• Analysis on the Nonlinear Dynamics of Deep Neural Networks: Topological Entropy and Chaos
• From Self-ception to Image Self-ception: A method to represent an image with its own approximations
• Who Asked Us? How the Theory of Computing Answers Questions about Analysis
• Hausdorff measure of cartesian product of Cantor sets
• Rectangular Shrinking Targets on Self-Similar Carpets
• On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks
• Exponents of inhomogeneous Diophantine Approximation
• Phase Transitions of the Multifractal Spectrum
• Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region
• On spectral Cantor measures
• Properties of a random Cantor set with overlaps
• Lower Assouad type dimensions of uniformly perfect sets in doubling metric spaces
• Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets
• Random walks associated to beta-shifts
• On totally disconnected generalised Sierpinski carpets
• On multifractal formalism for self-similar measures with overlaps
• One-dimensional quantum walks with one defect
• A study on Quantization Dimension in complete metric spaces
• Distinct dimensions for attractors of bi-Lipschitz iterated function systems
• Hausdorff Dimension of Cantor Series
• An ensemble of Gaussian fields with multifractal statistics for turbulence
• Asymptotic Divergences and Strong Dichotomy
• Uniform dimension results for the inverse images of symmetric Lévy processes
• Katugampola fractional integral and fractal dimension of bivariate functions
• Multivariate stable approximation in Wasserstein distance by Stein's method
• On the intermediate dimensions of Bedford-McMullen carpets
• Algorithmically Optimal Outer Measures
• Finding Dynamics for Fractals
• Similar and Self-similar Curves in Minkowski n-space
• Quantitative recurrence properties in conformal iterated function systems
• Left invertibility of discrete-time output-quantized systems: the linear\n case with finite inputs
• Fractal closures of geodesic planes in Hitchin manifolds
• On Assouad dimension and arithmetic progressions in sets defined by digit restrictions
• A Survey and Comparative Evaluation of Intrinsic Dimension Estimators under the Manifold Hypothesis
• The connection between Jackson and Hausdorff derivatives in the context of generalized statistical mechanics
• Metric theory of lower bounds on Weyl sums
• Box-counting measure of metric spaces
• Analogues of s-potential and s-energy of mass distribution on Cantor dyadic group and their relation to Hausdorff dimension
• The exact value of Hausdorff dimension of escaping sets of class B meromorphic functions
• Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator
• Ubiquity and a general logarithm law for geodesics
• Construction and box dimension of recurrent fractal interpolation surfaces
• On estimation of fractal dimension for 2D time-series based on functional relation between areas of covers
• Some Remarks on Hausdorff Measurability of Lipschitz Images in Metric Spaces
• Complexity and fractal dimensions for infinite sequences with positive entropy
• Calculating box dimension with the method of types
• Magnetization Dynamics in Quasiperiodic Magnonic Crystals
• Characteristic box dimension of unit-time map near nilpotent singularity of planar vector field and applications
• Quadrilateral Labyrinth Fractals
• From Nash to Cournot--Nash equilibria via Γ-convergence
• Convex minorant trees associated with Brownian paths and the continuum limit of the minimum spanning tree
• Microcanonical processing methodology for ECG and intracardial potential: application to atrial fibrillation
• Some results on Lower Assouad and quantization dimensions
• The Barrow entropies in the thermodynamics of high-dimensional Gauss-Bonnet black holes
• Dimension Of Inhomogeneous Sub-Self-Similar Sets
• Barnsley-Navascués fractal operators on Banach spaces on the Sierpiński gasket
• Ostrogradsky-Sierpiński-Pierce expansion: dynamical systems, probability theory and fractal geometry points of view
• Metric Theory for Continued Fractions with Multiple Large Partial Quotients
• The graph, range and level set singularity spectra of b-adic independent cascade function
• Metric Diophantine approximation and 'absolutely friendly' measures
• Inhomogeneous Diophantine Approximation on M₀-sets with restricted denominators
• Exact Hausdorff and packing measures of linear Cantor sets with overlaps
• Self-Similar Fractals and Arithmetic Dynamics
• Two spaces that already found their geometer in the thirties
• Multifractal analysis of Bernoulli convolutions associated with Salem numbers
• Toward the mechanics of fractal materials: mechanics of continuum with fractal metric
• Deep Nonparametric Regression on Approximate Manifolds: Non-Asymptotic Error Bounds with Polynomial Prefactors
• Hausdorff dimension and infinitesimal similitudes on complete metric spaces
• Describability via ubiquity and eutaxy in Diophantine approximation
• Electromagnetic space-time crystals. II. Fractal computational approach
• Some remarks on the exponential separation and dimension preserving approximation for sets and measures
• Derivative coordinates for analytic tree fractals and fractal engineering
• Fork and Join Queueing Networks with Heavy Tails: Scaling Dimension and Throughput Limit
• Discrete Fractal Dimensions of the Ranges of Random Walks in \Z^d Associate with Random Conductances
• Lipschitz equivalence of fractals and finite state automaton
• Generalised golden ratios over integer alphabets
• Dimer-monomer model on the generalized Tower of Hanoi graph
• A classification of Reifenberg properties
• Self-similar real trees defined as fixed-points and their geometric properties
• Level sets of prevalent Weierstrass functions
• The McMillan theorem for colored branching processes and dimensions of random fractals
• Hausdorff theory of dual approximation on planar curves
• Some new classes of directed graph IFSs
• Proteins analysed as virtual knots
• Box dimension of unit-time map near nilpotent singularity of planar vector field
• Shrinking Targets for Countable Markov Maps
• Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents
• Times two, three, five orbits on \mathbbT²
• Geometrical approach to modeling of nonlinear systems from experimental data
• Fourier dimension and avoidance of linear patterns
• Dimensions of C1-average conformal hyperbolic sets
• Lower S-Dimension of Fractal Sets
• The de Bruijn-Erdős theorem from a Hausdorff measure point of view
• Hausdorff dimension of the range and the graph of stable-like processes
• Example of a diffeomorphism for which the special ergodic theorem doesn't hold
• Generalized spaces of pointwise regularity: To a general framework for the WLM
• Dimension Extractors and Optimal Decompression
• Generalized s-convex function on fractal sets
• Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension
• Multifractal analysis of Lévy fields
• Every Sequence is Decompressible from a Random One
• On the ubiquity of the ruler sequence
• Modeling tensorial conductivity of particle suspension networks
• On the quasisymmetric minimality of homogeneous perfect sets
• Iteration of polynomial pair under Thue-Morse dynamic
• The Eigenvalue Problem for Linear and Affine Iterated Function Systems
• Multifractal analysis of weighted ergodic averages
• The random walk's guide to anomalous diffusion: a fractional dynamics approach
• On Bauschke-Bendit-Moursi modulus of averagedness and classifications of averaged nonexpansive operators
• Graph Hausdorff dimension, Kolmogorov complexity and construction of fractal graphs
• Hausdorff dimension of divergent diagonal geodesics on product of finite volume hyperbolic spaces
• Information, initial condition sensitivity and dimension in weakly chaotic dynamical systems
• Dataset-Adaptive Dimensionality Reduction
• On upper and lower fast Knintchine spectra of continued fractions
• Optimal parametrizations of a class of self-affine sets
• Stochastic 2-microlocal analysis
• Self-affine sets with non-compactly supported random perturbations
• The Multifractal Nature of Boltzmann Processes
• Threshold functions for substructures in random subsets of finite vector spaces
• Measure of Self-Affine Sets and Associated Densities
• Differentiability inside sets with upper Minkowski dimension one
• A note on pointwise convergence for the Schrödinger equation
• A numerical retro-action model relates rocky coast erosion to percolation theory
• Dirac operators and spectral triples for some fractal sets built on curves
• Circuit models for Sierpinski gasket antennas
• Nonlocal p-Laplacian evolution problems on graphs
• How the geometry of cities explains urban scaling laws and determines their exponents
• Fractal geometry of literature: first attempt to Shakespeare's works
• On the box dimension of graph of harmonic functions on the Sierpiński gasket
• On Besov regularity and local time of the stochastic heat equation
• Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets
• Pre-training without Natural Images
• How to prove that some Bernoulli convolution has the weak Gibbs property
• Exact uniform modulus of continuity and Chung's LIL for the generalized fractional Brownian motion
• Perturbation of self-similar sets and some regular configurations and comparison of fractals
• Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes
• On a mixed Khintchine problem in Diophantine approximation
• Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds
• Minimal Deterministic Echo State Networks Outperform Random Reservoirs in Learning Chaotic Dynamics
• Fourier transform and regularity of characteristic functions
• Physics Matters in PnP: Recovery Guarantees with the MMSE and NN Denoisers
• Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms
• Learning scale-variant features for robust iris authentication with deep learning based ensemble framework
• Open Problems in Computability Theory and Descriptive Set Theory
• Escaping Mode Collapse in LLM Generation via Geometric Regulation
• Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions
• Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure
• Infinite derivatives of the Takagi-Van der Waerden functions
• On the classification of fractal squares
• Dimension of invariant measures for affine iterated function systems
• On the projections of the multifractal Hewitt-Stromberg dimension functions
• Multifractal analysis of some multiple ergodic averages in linear Cookie-Cutter dynamical systems
• Spectral triples for AF C*-algebras and metrics on the Cantor set
• The discrepancy of (n_kx) with respect to certain probability measures
• Topological pressure of free semigroup actions for non-compact sets and Bowen's equation
• Non-linear Fokker-Planck equations from conformal metrics and scalar curvature
• Cubes, side lengths and centres
• Quantitative Hardy--Littlewood maximal inequalities and Wiener--Stein theorem on p.c.f. fractals
• Observation of vibrating systems at different time instants
• Reducible Iterated Graph Systems: multiscale-freeness and multifractals
• On fast Lyapunov spectra for Markov-Rényi maps
• Space of Ricci flows (II)
• Fractal dimensions of complex networks: advocating for a topological approach
• Dimension and natural parametrization for SLE curves
• Measures of the full Hausdorff dimension for a general Sierpiński carpet
• Existence, equivalence and spectrality of infinite convolutions in \R^d
• Finding Attractors of Continuous-Time Systems by Parameter Switching
• Self-Similarity in Geometry, Algebra and Arithmetic
• Multifractal phenomena and packing dimension
• Shape-aware Sampling Matters in the Modeling of Multi-Class Tubular Structures
• Fractal Dimension of the Kronecker Product Based Fractals
• Fractal Gaussian Networks: A sparse random graph model based on Gaussian Multiplicative Chaos
• Dimensions of random covering sets in Riemann manifolds
• Deterministic approximations of random reflectors
• Filled Julia and Mandelbrot Sets for Dynamics over the Normed Real Nonassociative Algebras
• Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
• Schmidt's theorem, Hausdorff measures and Slicing
• Fractal Dimension for Fractal Structures: A Hausdorff Approach
• Strictly self-similar fractals composed of star-polygons that are attractors of Iterated Function Systems
• Fractional Perimeter and Nonlocal Minimal Surfaces
• Approximation orders of real numbers by β-expansions
• Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces
• Cantor spectrum for multidimensional quasi-periodic Schrödinger operators
• Kakeya books and projections of Kakeya sets
• The Hausdorff dimension of the boundary of the Lévy dragon
• Hausdorff measure of sets of distributional chaotic pairs for shift maps
• Finding subsets of positive measure
• The fractal nature of an approximate prime counting function
• Diophantine approximation by orbits of Markov maps
• Absolute Value Constraint: The Reason for Invalid Performance Evaluation Results of Neural Network Models for Stock Price Prediction
• Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra
• A survey of average contractive iterated function systems
• Cyclic Cohomology Groups of Some Self-similar Sets
• Optimal Oracles for Point-to-Set Principles
• Core entropy and biaccessibility of quadratic polynomials
• Finite Point Configurations and the Regular Value Theorem in a Fractal setting
• Open dynamical systems with a moving hole
• A dimension drop phenomenon of fractal cubes
• Packing dimensions of the divergence points of self-similar measures with the open set condition
• Fractal Attractors in Random Nonlinear Iterated Function Systems: Existence, Stability, and Dimensional Properties
• Non-standard analysis for fractal calculus
• Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian
• Hausdorff dimension of frequency sets of univoque sequences
• On the asymptotic quantization error for the doubling measures on Moran sets
• The accuracy of roughness exponent measurement methods
• Sums of two dimensional spectral triples
• Bilateral Canonical Cascades: Multiplicative Refinement Paths to Wiener's and Variant Fractional Brownian Limits
• Properties of Bourbaki's Function
• A common generalization of infinite sum, unordered sum and integral
• Straight way to Thermo-Statistics, Phase Transitions, Second Law of\n Thermodynamics, but without Thermodynamic Limit
• Quantitative recurrence properties and homogeneous self-similar sets
• Hausdorff measure of escaping sets on certain meromorphic functions
• Second Law of Thermodynamics and Macroscopic Observables within Boltzmann's principle, an attempt
• The closed Steinhaus properties of σ-ideals on topological groups
• Zero-one law of Hausdorff dimensions of the recurrent sets
• On the intermediate dimensions of concentric spheres and related sets
• Arcwise connectedness of the boundaries of connected self-similar sets
• Algorithmic interpretations of fractal dimension
• Spectral Triples for the Variants of the Sierpinski Gasket
• whittlehurst: A Python package implementing Whittle's likelihood estimation of the Hurst exponent
• Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity
• Toward zeta functions and complex dimensions of multifractals
• Effectively closed sets of measures and randomness
• Graph fractal dimension and structure of fractal networks: a combinatorial perspective
• Multifractality of jump diffusion processes
• Singularity sets of Levy processes
• Modified Holder Exponents Approach to Prediction of the USA Stock Market Critical Points and Crashes
• Fractal Graph Contrastive Learning
• Generalized Convex Functions and Some Inequalities on Fractal Sets
• Local Hölder regularity for set-indexed processes
• Inhomogeneous Diophantine approximation on curves and Hausdorff dimension
• Spectral properties of a class of Sierpinski-type Moran measures on ℝⁿ
• On a Topological Problem of Strange Attractors
• Multivariate multifractal analysis of Levy functions. Part I: Determination of multifractal spectra
• Fractal analysis of Neimark-Sacker bifurcation
• Fractal Subseries of the Harmonic Series
• Dynamical Characterization of Fractal Objects: Determination of the Fine Fractal Topology Using the Energy Cover
• Estimating degrees of freedom in motor systems
• Hausdorff dimension of restricted Kakeya sets
• Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves
• Affine embeddings of Cantor sets and dimension of αβ-sets
• Measures and the Law of the Iterated Logarithm
• Fractional perimeter from a fractal perspective
• Packing dimension of images and graphs of Gaussian random fields with drift
• Irreversibility in a simple reversible model
• The Fractal Dimension of Projected Clouds
• A formula for the fractal dimension d approx. 0.87 of the Cantorian set underlying the Devil's staircase associated with the Circle Map
• Corrections to the running of gauge couplings due to quantum gravity
• Critical values for the β-transformation with a hole at 0
• Subordinators: Examples and Applications
• Dimension theoretical properties of generalized Baker's transformations
• Stochastic Models That Separate Fractal Dimension and the Hurst Effect
• On fractional parts of powers of real numbers close to 1
• Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms
• Evolution of critical correlations at the QCD phase transition
• Diophantine approximation by orbits of expanding Markov maps
• Multifractal Formalism and Inequality involving Packing Dimension
• Identifying differences in physical activity and autonomic function patterns between psychotic patients and controls over a long period of continuous monitoring using wearable sensors
• On uniform distribution of αβ-orbits
• Dimension theory of graphs and networks
• Wavelets for iterated function systems
• Dimension preserving approximation
• Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields
• On Bernstein classes of quasianalytic maps
• Richardson’s Pair Diffusion and the Stagnation Point Structure of Turbulence
• On optimal wavelet bases for the realization of microcanonical cascade processes
• Multimode AFM analysis of aluminum-doped zinc oxide thin films sputtered under various substrate temperatures for optoelectronic applications
• Coincidence and noncoincidence of dimensions in compact subsets of [0,1]
• Contour lines of the discrete scale-invariant rough surfaces
• Fractal Dimension as a measure of the scale of Homogeneity
• Fractal modes in unstable resonators
• Hausdorff Dimension, Its Properties, and Its Surprises
• The Correlation Dimension of Young Stars in Dwarf Galaxies
• Entropy rates and finite-state dimension
• A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures
• Random matrices, generalized zeta functions and self-similarity of zero distributions
• Affine embeddings and intersections of Cantor sets
• Ergodic Directions for Billiards in a Strip with Periodically Located Obstacles
• Revealing Latent Self-Similarity in Cellular Automata via Recursive Gradient Profiling
• On thin carpets for doubling measures
• A Fourier restriction theorem based on convolution powers
• Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions
• Fractal zeta functions and complex dimensions of relative fractal drums
• On the asymptotic magnitude of subsets of Euclidean space
• The Roughness of the Last Scattering Surface
• An exact algorithm for minimizing vertex guards on art galleries
• How many points have the same Engel and Sylvester expansions?
• A problem of Galambos on Engel expansions
• Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis
• Unbound Young Stellar Systems: Star Formation on the Loose
• Fractal squares with finitely many connected components *
• On the specification property and synchronization of uniqueq-expansions
• A useful lemma for calculating the Hausdorff dimension of certain sets in Engel expansions
• Entropy-Variance curves of binary sequences generated by random substitutions of constant length
• Digit expansions of numbers in different bases
• Self-similar vector measures of Markov-type operators
• Dimension rigidity in conformal structures
• Evolving power grids with self-organized intermittent strain releases: An analogy with sandpile models and earthquakes
• A generalized family of transcendental functions with one dimensional Julia sets
• NONLINEAR ANALYSIS OF PHYSIOLOGICAL SIGNALS: A REVIEW
• Left invertibility of I/O quantized linear systems in dimension 1: a number theoretic approach
• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems
• How likely can a point be in different Cantor sets
• Existence and Spectrality of random measures generated by infinite convolutions
• Lacunarity exponents
• Small unions of affine subspaces and skeletons via Baire category
• The ℓ^p-metrization of functors with finite supports
• Restrictions of Brownian motion
• On the Hausdorff dimension of microsets
• MFDFA: Efficient multifractal detrended fluctuation analysis in python
• The dimension of irregular set in parameter space
• Self-propulsion of droplets by spatially-varying surface topography
• Multiplicity of fixed points and growth of ε-neighborhoods of orbits
• Dimension structures
• Hausdorff dimension of the large values of Weyl sums
• THE FRACTAL DIMENSION OF STAR-FORMING REGIONS AT DIFFERENT SPATIAL SCALES IN M33
• Large intersection properties in Diophantine approximation and dynamical systems
• A bridge between Dubovitskiĭ–Federer theorems and the coarea formula
• Slide Statistics And Financial Returns
• From compression to compressed sensing
• Linear Fractional Stable Sheets: wavelet expansion and sample path properties
• The box-counting dimension of random box-like self-affine sets
• Riemannian-geometric entropy for measuring network complexity
• The sequence property for fractal dimensions
• Classical evolution of fractal measures on the lattice
• Dimension preserving set-valued approximation and decomposition via metric sum
• Set-Valued Fractal Approximation for Countable Data Sets
• Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness
• Some Analytical Properties of Multivariate Fractal Functions in Lebesgue Spaces
• Effects of mobility of small islands on growth in molecular-beam epitaxy
• Study of the Microstructure of Coal at Different Temperatures and Quantitative Fractal Characterization. [europepmc]

Related

• An Analysis of Fractal Features
• The Fractal Geometry of Nature
• Some questions related to fractals
• A simple and didactic method to calculate the fractal dimension - an interdisciplinary tool
• Fractal Calculus to Derive Fractal Frenet Equations for Fractal Curves