Science AI Journal
Mathematics · Research topic

Open research questions in Mathematical Dynamics and Fractals

85 unresolved questions extracted from the limitations and future-work sections of 329 Mathematical Dynamics and Fractals papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The paper identifies a gap in the understanding of the entropy of maps on the Gehman dendrite. 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

  • Characterizing the dynamics on the carrying simplex remains a significant challenge. The model's complexity and nonlinearity make it difficult to analyze.

    Global Dynamics of Three-Dimensional Lotka–Volterra Competition Models with Seasonal Succession: II. Uniqueness of Positive Fixed Points · 2026 · DOI

  • The complexity of population survival functions. The need for a universal transformation law. The difficulty of assessing adaptive capacity and reproductive success.

    Invariance of Population Survival Functions · 2026 · DOI

  • The study of chaotic dynamical systems has focused on the deterministic setting, leaving a gap in the understanding of random systems. Prior work has not provided a computable formula for the average measure-theoretic entropy of a family of expanding on average random Blaschke products.

    Average measure-theoretic entropy for a family of expanding on average random Blaschke products · 2026 · DOI

  • Future research can focus on generalizing the results to higher dimensions. Future research can focus on understanding the properties of ergodic systems. Future research can focus on applying the results to statistical physics and dynamical systems.

    Genericity of ergodicity for Sobolev homeomorphisms · 2026 · DOI

  • The paper identifies a gap in prior work on the genericity of ergodicity for Sobolev homeomorphisms. The paper identifies a gap in prior work on the properties of volume-preserving Sobolev homeomorphisms.

    Genericity of ergodicity for Sobolev homeomorphisms · 2026 · DOI

  • The lack of a general understanding of the homoclinic growth rate in higher-dimensional systems. The need for a quantitative understanding of the relationship between chaos and homoclinic orbits.

    Metric entropy and homoclinic growth rate · 2026 · DOI

  • The paper does not provide a comprehensive analysis of all types of fractal percolations. The results are limited to the specific variants of fractal percolation studied in the paper.

    Quasisymmetric mappings on two variants of fractal percolation · 2026 · DOI

  • Further analysis of quasisymmetric mappings on other types of fractal percolations. Investigation of the applications of quasisymmetric mappings in various fields.

    Quasisymmetric mappings on two variants of fractal percolation · 2026 · DOI

  • The lack of a multivariate version of the Auslander-Yorke dichotomy. The need for a characterization of the maximal equicontinuous factor by the regionally proximal relation.

    A note on multivariate diam-mean equicontinuity and frequent stability · 2026 · DOI

  • The paper faces the challenge of characterizing Nash manifolds with divisorial corners. It needs to address the issue of finding Nash functions arbitrarily close to zero with respect to the S µ topology.

    Nash approximation of differentiable semialgebraic maps · 2026 · DOI

  • The paper identifies a gap in the understanding of Nash approximation of differentiable semialgebraic maps. It addresses the need for a characterization of Nash manifolds with divisorial corners.

    Nash approximation of differentiable semialgebraic maps · 2026 · DOI

  • The lack of understanding of the interaction between nonresonance and regularity in ergodic theory. The absence of a weighted counterpart to Yoccoz-type results.

    Exponential convergence can happen in weighted Birkhoff averages via quasi‐periodicity with arbitrary nonresonance and low regularity · 2026 · DOI

  • The study focuses on C r diffeomorphisms with r > 1. The results are limited to compact Riemannian manifolds.

    Metric entropy and homoclinic growth rate · 2026 · DOI

  • Further study of multivariate diam-mean equicontinuity and frequent stability. Applications of the results to mathematical models of quasicrystals.

    A note on multivariate diam-mean equicontinuity and frequent stability · 2026 · DOI

  • The construction in the special case $ω(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $ω(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon > 0$.

    On the sharpness of Denjoy's theorem · 2026

  • Remark 7.3 notes that a hyperbolic Riemann surface X admits f ∈ Hol(X, X) with two distinct fixed points only if f = idX or X is multiply connected with f periodic; however, the behavior of left iterated function systems when the individual iterates fn have multiple fixed points and the effect of point selection strategy (closest to a in Theorem H proof) on convergence properties remains unexamined.

    Iterated function systems of holomorphic maps · 2026 · DOI

  • In Theorem I, the proof establishes that under the contraction condition ℓn ≤ ℓ < 1 on the derivative supremum, the left iterated function systems {Ln} converge pointwise to the fixed point a; however, the dependence of the convergence rate on the contraction ratio ℓ and initial conditions is not explicitly characterized.

    Iterated function systems of holomorphic maps · 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

  • The transition between positive Lyapunov exponent (Section 4.2) and negative Lyapunov exponent (Section 4.3) regimes involves different limiting behaviors of synchronized orbits, but the critical bifurcation point and the structure of intermittency precisely at λ = 0 are not rigorously characterized beyond the existence of zero Lyapunov exponent at the chaos transition.

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

  • The gap between the classical Banach-Hutchinson-Barnsley framework and the study of non-contractive IFSs. The need for a unified viewpoint for the study of IFSs beyond classical contractions.

    The Role of Point-Fibredness in Iterated Function Systems: Exposition, Exploration, and Extensions · 2026 · DOI

  • Nevertheless, whether EET extends to point-fibred or uniformly point-fibred IFSs remains an open problem and an appealing avenue for further investigation.

    The Role of Point-Fibredness in Iterated Function Systems: Exposition, Exploration, and Extensions · 2026 · DOI

Most-cited papers in Mathematical Dynamics and Fractals

Most recent work

Find a gap in your own Mathematical Dynamics and Fractals sub-topic

This page shows what the Mathematical Dynamics and Fractals literature already flags as unresolved. To narrow it to your specific question, run the guided finder — it searches the gap library on demand and checks candidates against 250M+ OpenAlex works.

Open the Research Gap Finder →

Related topics in Mathematics

Cognitive and developmental aspects of mathematical skills
398 open questions
Mathematics Education and Pedagogy
319 open questions
Statistics Education and Methodologies
146 open questions
Geometric Analysis and Curvature Flows
143 open questions
COVID-19 epidemiological studies
142 open questions
Advanced Causal Inference Techniques
140 open questions
Analytic Number Theory Research
118 open questions
Numerical methods for differential equations
113 open questions
85 open questions have been extracted from the limitations and future-work passages of 329 Mathematical Dynamics and Fractals papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

Tools for your next paper

Gaps & Literature SearchFind open questions in any field — 120,000+ open gaps mined from our own library.Pre-CheckIs your paper ready? A Tier 1–5 acceptance probability in seconds.Journal FinderA ranked shortlist of target journals from a 17,500-venue index.Duplicate Publication CheckerCheck prior-publication & salami-slicing risk across 8 sources.Citation GeneratorFreeA DOI or title → APA, MLA, Vancouver, Chicago, BibTeX, RIS. No signup.Graphical Abstract MakerFreeTurn your findings into a colorblind-safe graphical abstract.

Compare the category — Honest roundups of the AI research tools, ours listed alongside the alternatives.

Command palette

Jump anywhere, run any action.