Mathematics › Mathematical Physics

Mathematical Dynamics and Fractals

Works Count: 76268

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This cluster of papers covers a wide range of topics in dynamical systems and chaos theory, including chaos implications, ergodic theory, fractals, Lyapunov exponents, Markov chains, topological entropy, invariant measures, Teichmüller curves, and Hausdorff dimension.

Keywords

Chaos; Dynamical Systems; Fractals; Ergodic Theory; Lyapunov Exponents; Markov Chains; Topological Entropy; Invariant Measures; Teichmüller Curves; Hausdorff Dimension

Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

1975-01-01

Rufus Bowen

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Introduction to Compact Transformation Groups

1972-01-01

Glen E. Bredon

Recurrence in Ergodic Theory and Combinatorial Number Theory

1981-12-31

Harry Furstenberg

Topological dynamics and ergodic theory usually have been treated independently. H. Furstenberg, instead, develops the common ground between them by applying the modern theory of dynamical systems to combinatories and … Topological dynamics and ergodic theory usually have been treated independently. H. Furstenberg, instead, develops the common ground between them by applying the modern theory of dynamical systems to combinatories and number theory. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Detecting strange attractors in turbulence

1981-01-01

Floris Takens

Self-similarity and intermediate asymptotics

2003-11-13

Г. И. Баренблатт

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Random dynamical systems

2020-12-01

Ludwig Arnold

I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic … I. Random Dynamical Systems and Their Generators.- 1. Basic Definitions. Invariant Measures.- 2. Generation.- II. Multiplicative Ergodic Theory.- 3. The Multiplicative Ergodic Theorem in Euclidean Space.- 4. The Multiplicative Ergodic Theorem on Bundles and Manifolds.- 5. The MET for Related Linear and Affine RDS.- 6. RDS on Homogeneous Spaces of the General Linear Group.- III. Smooth Random Dynamical Systems.- 7. Invariant Manifolds.- 8. Normal Forms.- 9. Bifurcation Theory.- IV. Appendices.- Appendix A. Measurable Dynamical Systems.- A.1 Ergodic Theory.- A.2 Stochastic Processes and Dynamical Systems.- A.3 Stationary Processes.- A.4 Markov Processes.- Appendix B. Smooth Dynamical Systems.- B.1 Two-Parameter Flows on a Manifold.- B.4 Autonomous Case: Dynamical Systems.- B.5 Vector Fields and Flows on Manifolds.- References.

On the geometry and dynamics of diffeomorphisms of surfaces

1988-01-01

William P. Thurston

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An equation for continuous chaos

1976-07-01

Otto E. Rössler

An Introduction to Chaotic Dynamical Systems

1987-07-01

Robert L. Devaney , Jean‐Pierre Eckmann

Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural … Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

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Groups of Diffeomorphisms and the Motion of an Incompressible Fluid

1970-07-01

David G. Ebin , Jerrold E. Marsden

In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for … In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible fluid (both viscous and non-viscous) on a compact C^∞ riemannian, oriented n-manifold M, possibly with boundary.

Endomorphisms and automorphisms of the shift dynamical system

1969-12-01

Gustav A. Hedlund

CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY

1977-08-31

Ya. B. Pesin

CONTENTS Part I § 1. Introduction § 2. Prerequisites from ergodic theory § 3. Basic properties of the characteristic exponents of dynamical systems § 4. Properties of local stable manifolds … CONTENTS Part I § 1. Introduction § 2. Prerequisites from ergodic theory § 3. Basic properties of the characteristic exponents of dynamical systems § 4. Properties of local stable manifolds Part II § 5. The entropy of smooth dynamical systems § 6. "Measurable foliations". Description of the π-partition § 7. Ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure. The K-property § 8. The Bernoullian property § 9. Flows § 10. Geodesic flows on closed Riemannian manifolds without focal points References

An Introduction to Diophantine Approximation.

1958-06-01

John C. Brixey , J. W. S. Cassels

Chaos: An Introduction to Dynamical Systems

1997-11-01

Kathleen T. Alligood , Timothy Sauer , James A. Yorke +1 more

One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State … One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.

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GIBBS MEASURES IN ERGODIC THEORY

1972-08-31

Yakov G. Sinai

In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the … In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of Anosov dynamical systems. By means of this concept we construct a wide class of invariant measures for dynamical systems of this kind and investigate the problem of the existence of an invariant measure consistent with a smooth structure on the manifold; we also study the behaviour under small random excitations as . The cases of discrete time and continuous time are treated separately.

Fractal Geometry: Mathematical Foundations and Applications.

1990-09-01

K. J. Falconer

Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. … Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

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Groups of polynomial growth and expanding maps

1981-12-01

Mikhael Gromov

Mathematical problems for the next century

1998-03-01

Steve Smale

Asymptotic Theory for Econometricians.

1988-02-01

Peter M. Robinson , Hannah J. White

The Linear Model and Instrumental Variables Estimators. Consistency. Laws of Large Numbers. Asymptotic Normality. Central Limit Theory. Estimating Asymptotic Covariance Matrices. Functional Central Limit Theory and Applications. Directions for Further … The Linear Model and Instrumental Variables Estimators. Consistency. Laws of Large Numbers. Asymptotic Normality. Central Limit Theory. Estimating Asymptotic Covariance Matrices. Functional Central Limit Theory and Applications. Directions for Further Study. Solution Set. References. Index.

Persistence and Smoothness of Invariant Manifolds for Flows

1971-01-01

Neil Fenichel

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Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation

1967-03-01

Harry Furstenberg

Differentiable Dynamical Systems

1980-01-01

Steve Smale

Introduction to the modern theory of dynamical systems

1996-06-01

Anatole Katok , Boris Hasselblatt

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to … Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Dynamical systems with elastic reflections

1970-04-30

Yakov G. Sinai

In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second … In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. We prove that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.

A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems

1979-06-01

I. Shimada , Tomohiro Nagashima

Based on the Lyapunov characteristic exponents, the ergodic property of dissipative dynamical systems with a few degrees of freedom is studied numerically by employing, as an example, the Lorenz system. … Based on the Lyapunov characteristic exponents, the ergodic property of dissipative dynamical systems with a few degrees of freedom is studied numerically by employing, as an example, the Lorenz system. The Lorenz system shows the spectra of (+,0,-) type concerning the 1-dimensional Lyapunov exponents, and the exponents take the same values for orbits starting from almost of all initial points on the attractor.

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Convergence of Probability Measures

1970-02-01

James L. Snell

Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index. Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.

The infinite number of generalized dimensions of fractals and strange attractors

1983-09-01

H. G. E. Hentschel , Itamar Procaccia

Image coding based on a fractal theory of iterated contractive image transformations

1992-01-01

A. Jacquin

The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that (i) it relies … The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that (i) it relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis, and (ii) it approximates an original image by a fractal image. The author refers to the approach as fractal block coding. The coding-decoding system is based on the construction, for an original image to encode, of a specific image transformation-a fractal code-which, when iterated on any initial image, produces a sequence of images that converges to a fractal approximation of the original. It is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0.5-1.0 b/pixel. The fractal block coder has performance comparable to state-of-the-art vector quantizers.

Crises, sudden changes in chaotic attractors, and transient chaos

1983-05-01

Celso Grebogi , Edward Ott , James A. Yorke

Convergence of Probability Measures

1999-07-16

Patrick Billingsley

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An introduction to chaotic dynamical systems

1990-05-01

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Differentiable dynamical systems

1967-01-01

Stephen T. Smale

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An Introduction to Ergodic Theory

1982-01-01

Peter Walters

Embedology

1991-11-01

Timothy Sauer , James A. Yorke , Martin Casdagli

Asymptotics and Special Functions

1974-01-01

Asymptotics and Special Functions

1975-04-01

Wolfgang Wasow , F. W. Olver

Ergodic Theory

1982-01-01

I. P. Cornfeld , S. V. Fomin , Ya. G. Sinaĭ

Handbook of Brownian Motion - Facts and Formulae

2002-01-01

A. N. Borodin , Paavo Salminen

There are two parts in this book. The first part is devoted mainly to the proper ties of linear diffusions in general and Brownian motion in particular. The second part … There are two parts in this book. The first part is devoted mainly to the proper ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distribut

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Scale Mixtures of Normal Distributions

1974-09-01

David Andrews , C. L. Mallows

Summary This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio Z/V where Z and V are independent and Z has … Summary This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio Z/V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 1/2V 2 is exponential, X is double exponential; and that when 1/2V has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.

An Enhanced Fractal Image Compression Algorithm Based on Adaptive Non-Uniform Rectangular Partition

2025-06-24

M Li , Kin Tak U | Electronics

The Basic Fractal Image Compression (BFIC) method is widely known for its high computational complexity and long encoding time under a fixed block segmentation. To address these limitations, we propose … The Basic Fractal Image Compression (BFIC) method is widely known for its high computational complexity and long encoding time under a fixed block segmentation. To address these limitations, we propose an enhanced fractal image compression algorithm based on adaptive non-uniform rectangular partition (FICANRP). This novel approach adaptively partitions the image into variable-sized range blocks (R-blocks) and non-overlapping domain blocks (D-blocks) guided by local texture and feature. By converting the similarity-matching process for R-blocks into a localized search strategy based on block size and feature classification, the FICANRP method significantly reduces computational overhead. Moreover, employing a non-overlapping partition strategy for D-blocks drastically reduces the number of D-blocks and the associated spatial coordinate data while preserving high matching accuracy. This reduction, coupled with the block similarity matching algorithm that overcomes traditional fractal computation redundancy, significantly decreases algorithmic complexity and encoding time. Additionally, by adaptively segmenting R-blocks into varying sizes according to local texture, the proposed method minimizes redundancy in smooth regions while preserving fine details in complex areas. The experimental results show that compared with BFIC, FICANRP has a compression ratio (CR) improvement range of 0.84–2.29 times, a PSNR improvement range of 0.25–4.8 dB, and an acceleration encoding time efficiency improvement of 54.14×–1448.73×. Compared with QFIC, under the same PSNR, the FICANRP compression ratio (CR) improvement range is 0.87–19.12 times, and the accelerated encoding time (ET) efficiency is increased by 37.26×–114.83×.

DIMENSIONAL APPROXIMATION OF AN INHOMOGENEOUS ATTRACTOR WITHOUT ANY SEPARATION CONDITION

2025-06-23

Manuj Verma | Bulletin of the Australian Mathematical Society

Abstract We show that the Hausdorff dimension of the attractor of an inhomogeneous self-similar iterated function system (or self-similar IFS) can be well approximated by the Hausdorff dimension of the … Abstract We show that the Hausdorff dimension of the attractor of an inhomogeneous self-similar iterated function system (or self-similar IFS) can be well approximated by the Hausdorff dimension of the attractor of another inhomogeneous self-similar IFS satisfying the strong separation condition. We also determine a formula for the Hausdorff dimension of the algebraic product and sum of the inhomogeneous attractor.

A certain class of sofic-Dyck shifts and its orbit growth

2025-06-23

Azmeer Nordin , Mohd Salmi Md Noorani , Mohd Hafiz Mohd | Monatshefte für Mathematik

Multiscale topology and dynamic internal vectorial geometry - alpha group

2025-06-23

Cleber Souza Corrêa , Thiago Braido Nogueira de Melo | STUDIES IN ENGINEERING AND EXACT SCIENCES

This work sought to apply the Continuous Wavelet Transform (CWT) to the multiscale time-frequency analysis of non-stationary signals associated with the temporal evolution behavior of the complex eigenvalues of the … This work sought to apply the Continuous Wavelet Transform (CWT) to the multiscale time-frequency analysis of non-stationary signals associated with the temporal evolution behavior of the complex eigenvalues of the matrix M(θ(t)), with θ=2π radians. The results of CWT analysis have proven to be particularly effective in identifying oscillatory patterns, transient regimes and resonant features that are not evident through conventional spectral analysis, such as that which occurs using Fourier Transform. The results show that the CWT scalogram clearly reveals a persistent multiscale structure associated with an internal dynamic vector geometry, which exhibits a complex foliated topology. Thus, the use of CWT enabled a refined characterization of the system's spectral dynamics, serving as a key tool for identifying topological phenomena, internal resonances, and recurrent structures intrinsic to the vectorial geometry of the Alpha Group.

Some hyperbolicity revisited and robust transitivity

2025-06-23

Luis Pedro Piñeyrúa | Ergodic Theory and Dynamical Systems

Abstract In this article, we revisit the notion of some hyperbolicity introduced by Pujals and Sambarino [A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26 (1) … Abstract In this article, we revisit the notion of some hyperbolicity introduced by Pujals and Sambarino [A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26 (1) (2006), 281–289]. We present a more general definition that, in particular, can be applied to the symplectic context (something that was not possible for the previous one). As an application, we construct $C^1$ robustly transitive derived from Anosov diffeomorphisms with mixed behaviour on centre leaves.

Thickness, Toeplitz words, and orientation preserving Lipschitz classification of Moran sets*

2025-06-23

Liangyi Huang , Shishuang Liu , Hui Rao | Nonlinearity

Abstract We investigate the orientation preserving Lipschitz equivalence of fractal sets on the real line. First, we show that if two fractal sets have infinite asymptotic thickness, then any bi-Lipschitz … Abstract We investigate the orientation preserving Lipschitz equivalence of fractal sets on the real line. First, we show that if two fractal sets have infinite asymptotic thickness, then any bi-Lipschitz map between them must be orientation preserving. Secondly, for a class of homogeneous Moran sets with uniform patterns, we define Toeplitz substitutions according to the patterns; we show that two such Moran sets are Lipschitz equivalent if and only if one fixed point of the Toeplitz substitution is an essential lattice subsequence of that of the other one. Thirdly, as an application of the above results, we completely classified the Bedford–McMullen carpets with uniform fibres and Hausdorff dimension 1.

ON A SUBSET OF BAZILEVICH FUNCTIONS IDENTIFIED BY THE THREE-LEAF FUNCTION, MILLER-ROSS FUNCTION

2025-06-22

Ezekiel Abiodun Oyekan , Ayotunde Olajide Lasode , Oduyomi Michael Badejo | Journal of Mathematics Mechanics and Computer Science

A significant portion of the collection of analytic-univalent functions of the type $$ h(\zeta) = \zeta + \sum_{n=rm+1}^{\infty} a_n\zeta^n $$ whose definition is found in the unit disk $$\Omega:=\{z:|z|<1\},$$ is … A significant portion of the collection of analytic-univalent functions of the type $$ h(\zeta) = \zeta + \sum_{n=rm+1}^{\infty} a_n\zeta^n $$ whose definition is found in the unit disk $$\Omega:=\{z:|z|<1\},$$ is investigated in this work. Several subsets of the well-known set of Bazilevi\v{c} functions are included in this new set. The new set and its findings are developed using the Miller-Ross function, the Schwarz function, some multiplier operators, and some mathematical ideas such as subordination, set theory, infinite series generation, and convolution of some geometric expressions. Among the main achievements are the estimates for the coefficient bounds and the Fekete-Szeg\"o functional. Generally speaking, the new set reduces to a number of known subsets with some supposedly unique results when some parameters are altered inside their declaration intervals.

The Brownian loop measure on Riemann surfaces and applications to length spectra

2025-06-20

Yilin Wang , Yuhao Xue | Communications on Pure and Applied Mathematics

Abstract We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses … Abstract We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses the Brownian loop measure introduced by Lawler and Werner. In particular, we express the total mass of Brownian loops in a fixed free homotopy class on any Riemann surface in terms of the length of the geodesic representative for the complete constant curvature metric. This expression also allows us to write the electrical thickness of a compact set in separating 0 and , or the Velling–Kirillov Kähler potential, in terms of the Brownian loop measure and the zeta‐regularized determinant of Laplacian as a renormalization of the Brownian loop measure with respect to the length spectrum.

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Anosov flows and Liouville pairs in dimension three

2025-06-20

Thomas Massoni | Algebraic & Geometric Topology

On some properties of the local entropy of dynamical systems unwinded as a function of a point in the phase space

2025-06-20

А. Н. Ветохин | Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki

The article studies the nature of the dependence on the local entropy point of a non-autonomous dynamical system. It is proved that the local entropy is a function of the … The article studies the nature of the dependence on the local entropy point of a non-autonomous dynamical system. It is proved that the local entropy is a function of the second Baire class on the phase space, and its set of lower semicontinuity points forms an everywhere dense set of type $G_\delta$. An autonomous dynamical system is constructed such that the set of above semicontinuity points of the local entropy of this system is empty.

Scaled Packing Pressures on Subsets for Amenable Group Actions

2025-06-20

Zubiao Xiao , Hao Jia , Zhengyu Yin | Acta Mathematica Scientia

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Exploring Higher Order Local Entropy Functions via Baire Category Theory

2025-06-19

Tingting Wang , Bilel Selmi , Zhiming Li | Qualitative Theory of Dynamical Systems

Critical values for intermediate and box dimensions of projections and other images of compact sets

2025-06-18

Nicolas Angelini , Ursula Molter | Journal of Fractal Geometry Mathematics of Fractals and Related Topics

Given a compact set E \subset \mathbb{R}^{d} , we investigate for which values of m the equality \operatorname{dim}_{\theta} P_{V}(E) = m or \operatorname{dim}_{\theta} P_{V}(E) = \operatorname{dim}_{\theta} E holds for \gamma_{d,m} … Given a compact set E \subset \mathbb{R}^{d} , we investigate for which values of m the equality \operatorname{dim}_{\theta} P_{V}(E) = m or \operatorname{dim}_{\theta} P_{V}(E) = \operatorname{dim}_{\theta} E holds for \gamma_{d,m} -almost all V \in G(d,m) . Our result extends to more general functions, including orthogonal projections and fractional Brownian motion. As a particular case, when \theta = 1 , the results apply to the box dimension.

Topological structure of isolated points in the space of $\mathbb {Z}^d$ -shifts

2025-06-18

Silvère Gangloff , Alonso Núñez | Ergodic Theory and Dynamical Systems

Abstract R. Pavlov and S. Schmieding [On the structure of generic subshifts. Nonlinearity 36 (2023), 4904–4953] recently provided some results about generic $\mathbb {Z}$ -shifts, which rely mainly on an … Abstract R. Pavlov and S. Schmieding [On the structure of generic subshifts. Nonlinearity 36 (2023), 4904–4953] recently provided some results about generic $\mathbb {Z}$ -shifts, which rely mainly on an original theorem stating that isolated points form a residual set in the space of $\mathbb {Z}$ -shifts such that all other residual sets must contain it. As a direction for further research, they pointed towards genericity in the space of $\mathbb {G}$ -shifts, where $\mathbb {G}$ is a finitely generated group. In the present text, we approach this for the case of $\mathbb {Z}^d$ -shifts, where $d \ge 2$ . As it is usual, multidimensional dynamical systems are much more difficult to understand. In light of the result of R. Pavlov and S. Schmieding, it is natural to begin with a better understanding of isolated points. We prove here a characterization of such points in the space of $\mathbb {Z}^d$ -shifts, in terms of the natural notion of maximal subsystems that we also introduce in this article. From this characterization, we recover the result of R. Pavlov and S. Schmieding for $\mathbb {Z}^1$ -shifts. We also prove a series of results that exploit this notion. In particular, some transitivity-like properties can be related to the number of maximal subsystems. Furthermore, we show that the Cantor–Bendixon rank of the space of $\mathbb {Z}^d$ -shifts is infinite for $d>1$ , while for $d=1$ , it is known to be equal to one.

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Sharp rates of convergence in the Hausdorff metric for some compactly supported stationary sequences

2025-06-17

Sana Louhichi | Journal of Applied Probability

Abstract We study convergence rates, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact … Abstract We study convergence rates, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact subset of $\mathbb{R}^d$ . We propose two different approaches for this study. The first is based on the notion of a minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index, which is much used in extreme value theory. The second approach is based on a $\beta$ -mixing condition together with a local-type dependence assumption. More precisely, all our results concern stationary $\beta$ -mixing sequences satisfying a tail condition, known as the ( a , b )-standard assumption, together with a local-type dependence condition or stationary sequences satisfying the ( a , b )-standard assumption and having a positive minimal index. We prove that the optimal rates of the independent and identically distributed setting can be reached. We apply our results to stationary Markov chains on a ball, or to a class of Markov chains on a circle or on a torus. We study with simulations the particular examples of a Möbius Markov chain on the unit circle and of a Markov chain on the unit square wrapped on a torus.

Baby Mandelbrot Sets and Spines in Some One-Dimensional Subspaces of the Parameter Space for Generalized McMullen Maps

2025-06-17

Suzanne Boyd , Matthew Hoeppner | Qualitative Theory of Dynamical Systems

None

2025-06-17

François Maucourant | Annales de l’institut Fourier

The main result of this paper is an analogue for a continuous family of tori of Kronecker–Weyl’s unique ergodicity of irrational rotations. We show that the notion corresponding in this … The main result of this paper is an analogue for a continuous family of tori of Kronecker–Weyl’s unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is satisfied in some homogeneous dynamical systems. This is used to prove the ergodicity of naturals lifts of invariant measures.

None

2025-06-17

Hideki Miyachi , Ken’ichi Ohshika , Athanase Papadopoulos | Annales de l’institut Fourier

In this paper, we introduce a new asymmetric weak metric on the Teichmüller space of a closed orientable surface with (possibly empty) punctures. This new metric, which we call the … In this paper, we introduce a new asymmetric weak metric on the Teichmüller space of a closed orientable surface with (possibly empty) punctures. This new metric, which we call the Teichmüller–Randers metric, is an asymmetric deformation of the Teichmüller metric and is obtained by adding to the infinitesimal form of the Teichmüller metric a differential 1-form. We study basic properties of the Teichmüller–Randers metric. In the case when the 1-form is exact, any Teichmüller geodesic between two points is also a unique Teichmüller–Randers geodesic between them. A particularly interesting case is when the differential 1-form is the differential of the logarithm of the extremal length function associated with a measured foliation. We show that in this case the Teichmüller–Randers metric is incomplete in any Teichmüller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.

Dynamics of nearly abelian transcendental semigroup

2025-06-16

Ramanpreet Kaur , D. Ramesh Kumar | The Journal of Analysis

Periodic minimum in the count of binomial coefficients not divisible by a prime

2025-06-16

Hsien‐Kuei Hwang , Svante Janson , Tsung‐Hsi Tsai | Mathematics of Computation

The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the … The summatory function of the number of binomial coefficients not divisible by a prime is known to exhibit regular periodic oscillations, yet identifying the less regularly behaved minimum of the underlying periodic functions has been open for almost all cases. We propose an approach to identify such minimum in some generality, solving particularly a previous conjecture of B. Wilson [Acta Arith. 83 (1998), 105–116].

On the fractal properties of generalized Cantor sets and Devil’s staircase functions

2025-06-16

Thomas Devos , Laurent Loosveldt , Samuel Nicolay | European Journal of Mathematics

On the Parameter Space of Fibered Hyperbolic Polynomials

2025-06-16

Robert Florido , Núria Fagella | Journal of Geometric Analysis

The uniform Gardner conjecture and rounding Borel flows

2025-06-16

Matthew L. Bowen , Gábor Kun , Marcin Sabok | Proceedings of the American Mathematical Society

We study groups which satisfy Gardner’s equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit <inline-formula … We study groups which satisfy Gardner’s equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper Z slash 2 double-struck upper Z right-parenthesis asterisk left-parenthesis double-struck upper Z slash 2 double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∗</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {Z}/ 2\mathbb {Z}) * (\mathbb {Z}/ 2\mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a quotient by a finite subgroup. One of our technical contributions is an algorithm for rounding Borel flows for actions of amenable groups.

Stable functions and Følner’s Theorem

2025-06-16

Gabriel Conant | Proceedings of the American Mathematical Society

We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an amenable group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to upper … We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an amenable group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A subset-of-or-equal-to upper G"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\subseteq G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has positive upper Banach density, then there is an identity neighborhood <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Bohr topology on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is almost contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper A Superscript hyphen 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>-</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">AA^{\text {-}1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the sense that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B minus upper A upper A Superscript hyphen 1"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>-</mml:mtext> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">B\backslash AA^{\text {-}1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has upper Banach density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This generalizes the abelian case (due to Følner) and the countable case (due to Beiglböck, Bergelson, and Fish). The proof is indirectly based on local stable group theory in continuous logic. The main ingredients are Grothendieck’s double-limit characterization of relatively weakly compact sets in spaces of continuous functions, along with results of Ellis and Nerurkar on the topological dynamics of weakly almost periodic flows.

Polynomial Rate of Mixing for the Heterochaos Baker Maps with Mostly Neutral Center

2025-06-16

Hiroki Takahasi , Masato Tsujii | Journal of Dynamics and Differential Equations

The fractal uncertainty principle via Dolgopyat’s method in higher dimensions

2025-06-13

Aidan Backus , James Leng , Zhongkai Tao | Analysis & PDE

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Kripke Attractors

2025-06-13

T. D. P. Brunet | Erkenntnis

Abstract Could we prevent the death of a friend by travelling in time? Claims like this are usually settled by considering the structure of time: we could prevent their death … Abstract Could we prevent the death of a friend by travelling in time? Claims like this are usually settled by considering the structure of time: we could prevent their death provided time branches; we could not prevent their death if time is linear. Branching time allows changing timelines, and on some branches we might prevent their death. However, what would we say about the structure of time if, despite efforts under time-travel and repeated changes to the past, we continually failed to save our friend? In many time-travel fictions, certain events robustly reoccur across different branches. Some propositions, deaths especially, have an attraction despite changes to the past. In this essay I apply the notion of attractor from dynamical system theory to the set of possible worlds used to interpret a proposition within a Kripke model. A proposition is an attractor if there is a region of the model where the accessible worlds lead invariably to the extension of the proposition. Accessibility relations can have inevitable asymptotic structure. I argue that treating a proposition as an attractor in a Kripke model is a good way to provide an account of the inevitability of events.

A model of the cubic connectedness locus

2025-06-12

Alexander Blokh , Lex Oversteegen , Vladlen Timorin +1 more | Nonlinearity

Abstract We describe a locally connected model of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of … Abstract We describe a locally connected model of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This model is similar to a quotient of the combinatorial quadratic Mandelbrot set in which all baby Mandelbrot sets, as well as the filled Main Cardioid, are collapsed to points. All fibres of the model, possibly except one, are connected. The authors are not aware of other known models of the cubic connectedness locus.

Quasi-uniform entropy under free semigroup actions on quasi-uniform spaces

2025-06-12

Yang Zhong-xuan , Jimmy Xiangji Huang | Dynamical Systems

Variational principles for neutralized BS-dimension of subsets

2025-06-11

Yang Zhong-xuan | Dynamical Systems

On Dirichlet non-improvable numbers and shrinking target problems

2025-06-11

Qian Xiao | Ergodic Theory and Dynamical Systems

Abstract In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh … Abstract In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [ Proc. Amer. Math. Soc. 146 (5) (2018), 1833–1844] made a seminal contribution by linking the improvability of Dirichlet’s theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B> 1$ . We determine the Hausdorff dimension of the following set: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

Fractal-induced flow dynamics: Viscous flow around Mandelbrot and Julia sets

2025-06-09

Mutaz Mohammad , Alexander Trounev | Chaos Solitons & Fractals

Minimal Flow Morsification subject to Level Set Control: a Combinatorial Approach

2023-11-06

Maria Alice Bertolim , Ketty A. de Rezende , Margarida Pinheiro Mello

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