Open research questions in Mathematical Dynamics and Fractals

What the literature leaves open

• The core theoretical contribution is not the re-proof of the positivity of Lyapunov exponents—which directly follows from the robust chaos of the underlying skew map—but rather the formalization of a fuzzy framework that translates such robustness into the domain of parametric uncertainty.

  Fuzzy Skew Maps: Preserving Robust Chaos Under Uncertainty with Applications to Cryptography · 2026 · DOI

• The need for a novel technique to generate and evaluate Julia and Mandelbrot sets. The lack of a refined procedure to enable the generation of novel Mandelbrot and Julia sets.

  Fractal Analysis via Extended Fibonacci-Mann Iteration · 2026 · DOI

• The stationary measure construction in Section 5 via inducing methods is stated to apply to zero and positive Lyapunov exponent cases, but the explicit form of the stationary measure density and its asymptotics as points approach the diagonal Ω are deferred to later analysis, and the measure's absolute continuity properties relative to the product measure on T² \ Ω remain to be determined.

  Intermittent Two-Point Dynamics at the Transition to Chaos for Random Circle Endomorphisms · 2026 · DOI

• Further research can be done on the applications of the results to image processing and data compression. Further research can be done on the extension of the results to other areas of mathematics.

  Fractal modeling with coupled iterated function systems · 2026 · DOI

• The gap is related to the existence of a unique coupled best attractor for two distinct classes of contractions.

  Fractal modeling with coupled iterated function systems · 2026 · DOI

• Future research can build on the results of the paper to study non-hyperbolic ergodic measures in other contexts. Future research can apply the techniques of the paper to other areas of dynamical systems.

  Zero Lyapunov exponents in transitive skew products of iterated function systems · 2026 · DOI

• The gap is the lack of understanding of non-hyperbolic ergodic measures in transitive skew products. The gap is the need for a robust criterion for the existence of nonhyperbolic ergodic measures.

  Zero Lyapunov exponents in transitive skew products of iterated function systems · 2026 · DOI

• The authors aim to address this gap by constructing a continuous pure mixing map with topological entropy α for every 0 < α ≤ ∞.

  On entropy of pure mixing maps on dendrites · 2026 · DOI

• The action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation is complex. The mixing properties of this action are not well understood. The paper needs to extend some results on random walks of Abbott and the first author.

  Subgroup mixing and random walks in groups acting on hyperbolic spaces · 2026 · DOI

• There is a need to understand the mixing properties of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. The prior work does not provide a complete understanding of this action.

  Subgroup mixing and random walks in groups acting on hyperbolic spaces · 2026 · DOI

• The lack of understanding of mixing flows on finite-area translation surfaces. The need for a positive answer to a question posed by Lindsey and Treviño.

  On mixing flows on finite-area translation surfaces · 2026 · DOI

• Future research could focus on generalizing the findings of this paper to other families of maps.

  Strange bifurcation diagrams · 2026

• There is a gap in the understanding of the behavior of the EOS100 maps and their bifurcation diagrams. The paper identifies a need for further study of these maps and their properties. The authors note that the EOS100 maps are a different family of maps that has not been extensively studied.

  Strange bifurcation diagrams · 2026

• The GBLP measure can be unfaithful in certain cases. The BLP measure is unaffected by the translation vector ⃗ c(t). The analysis is limited to the study of complete positive channels.

  On the distinction between distinguishability of states and witness of non-Markovianity of dynamical maps · 2026 · DOI

• The distinction between distinguishability of states and witness of non-Markovianity of dynamical maps is not well understood. The limitations of the GBLP measure are not well understood.

  On the distinction between distinguishability of states and witness of non-Markovianity of dynamical maps · 2026 · DOI

• The literature lacks a complete analytical description of the dynamical behaviour of spatially-discretised quadratic maps. Prior work has focused on general theories and specific cases, but a detailed analysis of the map's behaviour for all values of $\lambda$ is lacking.

  The behaviour under iteration of a class of spatially-discretised quadratic maps · 2026 · DOI

• The gap in prior work is the lack of a general result for the Hausdorff dimension of random subsets of self-similar sets. The gap is addressed by this paper.

  The dimension of random subsets of self-similar sets generated by branching random walk · 2026 · DOI

• The paper does not provide a general formula for the barcode entropy. The results are limited to a specific class of invariant measures. The paper does not discuss the computational complexity of the methods used.

  From barcode entropy to metric entropy · 2026 · DOI

• The paper identifies a need for a new technique to study pillowcase-tiled surfaces.

  An infinite family of non-cyclic 1-cylinder pillowcase-tiled surfaces · 2026 · DOI

• The corresponding transition in random dynamical systems remains much less understood. The question of when and why chaotic dynamics emerges from more predictable motion has been a central question in bifurcation theory.

  Intermittent Two-Point Dynamics at the Transition to Chaos for Random Circle Endomorphisms · 2026 · DOI

• The paper notes that prior work has not provided a criterion for the weak Gibbs property of equilibrium measures for (κ/β, β)-shifts.

  Weak Gibbs measures for the natural extension of (κ/β, β)-shifts · 2026 · DOI

• The study suggests extending the work to cover more transcendental functions. The study suggests exploring the potential applications of the Picard-Thakur iteration method.

  Julia and Mandelbrot Sets of Transcendental Cosine-Function Using Picard-Thakur Iteration Method · 2026 · DOI

• The study aims to investigate how parameter variations influence the topology of fractal structures.

  Julia and Mandelbrot Sets of Transcendental Cosine-Function Using Picard-Thakur Iteration Method · 2026 · DOI

• Further study of iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Further study of the applications of the paper's results to work on transcendental dynamics and the theory of hyperbolic steps of holomorphic maps.

  Iterated function systems of holomorphic maps · 2026 · DOI

• To study the computational complexity of the methods used. To generalize the results to a broader class of invariant measures. To apply the results to other areas of study, such as the study of hyperbolic sets.

  From barcode entropy to metric entropy · 2026 · DOI

Most-cited papers in Mathematical Dynamics and Fractals

• Period of a Discrete Cat Mapping · American Mathematical Monthly · 1992 · 197 citations
• Dislocating identity: Desegregation and the transformation of place · Journal of Environmental Psychology · 2004 · 168 citations
• Dimension of a quantum-mechanical path · American Journal of Physics · 1981 · 146 citations
• Towards a general theory of topological maps · Artificial Intelligence · 2003 · 141 citations
• The Game of Hex and the Brouwer Fixed-Point Theorem · American Mathematical Monthly · 1979 · 137 citations
• Topographic mapping of the spectral components of the cyclic alternating pattern (CAP) · Sleep Medicine · 2004 · 107 citations
• Choropleth Maps Without Class Intervals? · Geographical Analysis · 1973 · 95 citations
• The effect of image variations on fractal dimension calculations · Oral Surgery Oral Medicine Oral Pathology Oral Radiology and Endodontology · 1997 · 58 citations
• Chaos, Fractals and Statistics · Statistical Science · 1992 · 55 citations
• The Conley index and the critical groups via an extension of Gromoll-Meyer theory · Topological Methods in Nonlinear Analysis · 1996 · 46 citations

Most recent work

• Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms · Advances in Mathematics · 2026
• Topological Stability of Semigroup Actions and Shadowing · Journal of Dynamical and Control Systems · 2026
• On the cuspy structure of rotating wormhole shadows · The European Physical Journal C · 2026
• Polynomial ergodic theorems in the spirit of Dunford and Zygmund · Advances in Mathematics · 2026
• Divisibility of Dynamical Maps: Schrödinger Versus Heisenberg Picture · PRX Quantum · 2026
• REMARKS ON THE INVARIANCE THEOREM OF GENERALIZED FRACTAL DIMENSIONS OF GRAPHS OF CONTINUOUS FUNCTIONS · Fractals · 2026
• Analysis of Julia sets using viscosity approximation-based iteration processes · Numerical Algorithms · 2026
• Li-Yorke chaos on fuzzy dynamical systems · RiuNet (Universitat Politècnica de València) · 2026
• Topological entropy for non-autonomous iterated function systems · Journal of Mathematical Analysis and Applications · 2026
• Discrete fractals: Dimensions, quasi-isometric invariance and self-similarity · Advances in Mathematics · 2026

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