Daily Dynamics & Continua DigestHigh-priority and related new arXiv papers in math.DS and math.GNhttps://dyntopo.pages.dev/2026-10-09T08:28:59.957671ZClaude[HIGH] Local Connectivity of the Filled Julia Set Bundle over the Mandelbrot Sethttps://dyntopo.pages.dev/#2026-10-09/2610.117932026-10-09T08:28:59.957671ZYueyang WangThe non-escaping locus of the quadratic family over the Mandelbrot set has boundary of full Hausdorff dimension 4 and is locally connected at hyperbolic parameters, but local connectedness fails at some real parabolic parameters. [complex dynamics, continuum theory][Related] The Milnor Fiber as a Local Conley Indexhttps://dyntopo.pages.dev/#2026-10-09/2610.108962026-10-09T08:28:59.957671ZJean-Paul Brasselet, Nivaldo GrulhaFor a holomorphic germ with an isolated critical point, the origin is an isolated invariant set of the negative gradient flow of \(\operatorname{Re} f\), with Conley index the suspension of the Milnor fibre. [flows and foliations][Related] A Universal Quasi-Polish Spacehttps://dyntopo.pages.dev/#2026-10-09/2610.114172026-10-09T08:28:59.957671ZSu Gao, Qingguo Li, Hualin MiaoThere is a universal quasi-Polish space containing every quasi-Polish space as a closed subspace, and in its hyperspace the Polish closed subspaces form a complete coanalytic set. [hyperspaces and descriptive topology][Related] Continuity of the Julia sets in non-archimedean and hybrid settingshttps://dyntopo.pages.dev/#2026-10-09/2610.115452026-10-09T08:28:59.957671ZAlexandre RoyAnalogues of the Mañé–Sad–Sullivan continuity of Julia sets in the Hausdorff topology hold for families of rational maps parametrized by Berkovich spaces and in the hybrid setting. [complex dynamics, hyperspaces and descriptive topology][Related] Variational Principles for $r$-Neutralized Pfister-Sullivan Entropyhttps://dyntopo.pages.dev/#2026-10-09/2610.116392026-10-09T08:28:59.957671ZErcai Chen, Yunxiang Xie, Xiaoyao ZhouThe authors disprove a variational conjecture of Dong and Qiao for r-neutralized entropy and establish variational formulas and bounds for r-neutralized Pfister–Sullivan entropy. [entropy and chaos][Related] Intersections of quantitative recurrence and badly approximable sets for expanding Markov mapshttps://dyntopo.pages.dev/#2026-10-09/2610.122582026-10-09T08:28:59.957671ZLingmin Liao, Na YuanFor expanding Markov interval maps, the Hausdorff dimension of recurrence sets intersected with dynamically badly approximable sets is given by the same pressure function as for the recurrence sets. [interval and graph maps, fractals and dimension][Related] Foliations forcing closed orbitshttps://dyntopo.pages.dev/#2026-10-09/2610.105582026-10-09T08:28:59.957671ZEllis Buckminster, Audrey RosevearThe authors characterize which finite depth foliations on atoroidal 3-manifolds force every transverse flow to have a closed orbit, and show this can often be broken by \(C^0\)-small changes. [flows and foliations][Related] Invariant sets and spectral rigidity of Hamiltonian diffeomorphismshttps://dyntopo.pages.dev/#2026-10-09/2610.122792026-10-09T08:28:59.957671ZHabib Alizadeh, Egor ShelukhinEvery non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterates separated from the identity in Hofer's metric, answering a question of Polterovich. [planar and surface dynamics][Related] Drilling veering triangulations with applications to pseudo-Anosov flowshttps://dyntopo.pages.dev/#2026-10-09/2610.123022026-10-09T08:28:59.957671ZAnna Parlak, Henry SegermanAn algorithm drills closed orbits out of transitive pseudo-Anosov flows at the level of veering triangulations, giving computational evidence for a conjecture of Ghys. [flows and foliations, smooth and hyperbolic dynamics][HIGH] Typical expansive measures for infinite-dimensional mapshttps://dyntopo.pages.dev/#2026-10-08/2610.094822026-10-09T08:27:35.973200ZGabriel LacerdaThe shift on the Hilbert cube is expansive with respect to product Lebesgue measure, so Mañé's finite-dimensionality theorem for expansive homeomorphisms does not extend to measure expansiveness. [genericity, shadowing and specification][Related] Ergodic measures of intermediate entropies for amenable group actions with an approximate product propertyhttps://dyntopo.pages.dev/#2026-10-08/2610.099492026-10-09T08:27:35.973200ZWenda Zhang, Xiankun RenAn approximate product property for amenable group actions yields entropy-denseness, and with asymptotic entropy expansiveness every entropy in \([0,h(X,G))\) is realized by a residual set of ergodic measures. [entropy and chaos, group actions, shadowing and specification][Related] Universal Dynamics in a family of Celestial Mechanics modelshttps://dyntopo.pages.dev/#2026-10-08/2610.099662026-10-09T08:27:35.973200ZMiguel Garrido, Pau Mart\'inEvery symplectic embedding of the disk into the plane is approximated by a renormalized Poincaré map of a suitable restricted planar circular many-body problem. [celestial mechanics][Related] Non-rigid disk pseudo-rotations in $C^1$ and $C^{1,\beta}$https://dyntopo.pages.dev/#2026-10-08/2610.099862026-10-09T08:27:35.973200ZZiran Liu, Jian Wang, Peizheng YuThe authors construct area-preserving irrational pseudo-rotations of the closed disk that are not \(C^0\)-rigid, arbitrarily close to the rigid rotation in \(C^1\) or \(C^{1,\beta}\). [planar and surface dynamics][Related] Variational Persistence of Periodic and Quasi-Periodic Orbits in Symmetric $N$-Body Problemshttps://dyntopo.pages.dev/#2026-10-08/2610.101012026-10-09T08:27:35.973200ZHangqi Liang, Zhihong Xia, Tingjie ZhouRotational symmetry yields families of periodic and quasi-periodic N-body trajectories via variational persistence results. [celestial mechanics][Related] The regionally proximal relation for commutative semigroup actionshttps://dyntopo.pages.dev/#2026-10-08/2610.103972026-10-09T08:27:35.973200ZAngelina Blahodatna, Lauren Detmold, Daniel Glasscock, Anh N. LeFor minimal actions of commutative semigroups, the regionally proximal and equicontinuous structure relations are equivalence relations and coincide. [minimal systems and factors, group actions][Related] The local dynamical structure of $\Delta^*$ sets via a new Furstenberg family algebrahttps://dyntopo.pages.dev/#2026-10-08/2610.104162026-10-09T08:27:35.973200ZAngelina Blahodatna, Lauren Detmold, Daniel Glasscock, Anh N. LeSets meeting every difference subset of a commutative semigroup have local Bohr structure. [minimal systems and factors][Related] Hyperbolicity and obstructions to PL actions of the circlehttps://dyntopo.pages.dev/#2026-10-08/2608.121682026-10-09T08:27:35.973200ZLeonardo Dinamarca, Maximiliano Escayola, Sang-hyun Kim, Thomas KoberdaFinitely generated groups of PL circle homeomorphisms satisfy a trichotomy, giving a free-or-surface alternative for word-hyperbolic such groups. [group actions, circle and one-dimensional dynamics][HIGH] Period spectrum and rigidity of interval mapshttps://dyntopo.pages.dev/#2026-10-07/2610.074352026-10-09T08:26:50.666775ZGabriel Fuhrmann, Maik Gr\"oger, Alejandro PasseggiPeriodic orbits determine every topologically mixing interval map. For non-mixing maps with dense periodic points they determine the second iterate, and locally they determine an iterate on each basic set. [interval and graph maps, mixing and transitivity][HIGH] Shadowing and Topological Stability via Generalized Pseudo-Hyperbolicity in Autonomous and Nonautonomous Linear Dynamicshttps://dyntopo.pages.dev/#2026-10-07/2610.084262026-10-09T08:26:50.666775ZDavor Dragicevic, Mihaly PitukFor sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to shadowing and to topological stability. In particular shadowing and topological stability coincide for single operators. [shadowing and specification][HIGH] Classification complexity of homeomorphism group actionshttps://dyntopo.pages.dev/#2026-10-07/2608.303462026-10-09T08:26:50.666775ZMichal Hevessy, Benjamin VejnarReplacing the full homeomorphism group of a compact metrizable space by natural dense non-closed subgroups can strictly increase, or make incomparable, the complexity of the associated orbit equivalence relations. [hyperspaces and descriptive topology, group actions][Related] Rudolph-Johnson without dynamics: endomorphic rigidity on the circlehttps://dyntopo.pages.dev/#2026-10-07/2610.073172026-10-09T08:26:50.666775ZPeter Burton, Kate JuschenkoEntropy alone, without invariance, pushes a measure on the circle toward Lebesgue measure under any sublacunary set of maps \(x\mapsto mx\). [ergodic theory, entropy and chaos, circle and one-dimensional dynamics][Related] Mixing of the BCZ Maphttps://dyntopo.pages.dev/#2026-10-07/2610.073472026-10-09T08:26:50.666775ZAlbert Artiles, Andrew Best, Matteo BordignonThe Boca–Cobeli–Zaharescu map is strongly mixing with respect to normalized Lebesgue measure on the BCZ triangle. [ergodic theory, mixing and transitivity][Related] Groupoid strict comparison for Ample Groupoids: Fiberwise Supramenability and Topological Amenabilityhttps://dyntopo.pages.dev/#2026-10-07/2610.085112026-10-09T08:26:50.666775ZChunlin Liu, Xin Ma, Jianchao WuFiberwise supramenability or topological amenability of a minimal ample groupoid gives an almost unperforated clopen type semigroup, hence groupoid strict comparison in the sigma-compact case. [group actions, minimal systems and factors][Related] An Entropy Bound for Mean Dimension of Open Covershttps://dyntopo.pages.dev/#2026-10-07/2610.086252026-10-09T08:26:50.666775ZTal BarakThe mean dimension of a finite open cover is bounded in terms of its topological entropy and cardinality, so mean dimension pairs are entropy pairs and uniformly positive mean dimension implies uniformly positive entropy. [entropy and chaos, group actions][HIGH] Discontinuous Generalized Hyperbolicity and Structural Stabilityhttps://dyntopo.pages.dev/#2026-10-06/2610.041792026-10-09T08:26:07.415775ZSergey TikhomirovA continuous-selection method gives semiconjugacies and structural stability for Banach-space dynamics with possibly discontinuous generalized-hyperbolic splittings. It also yields Lipschitz shadowing for an induced operator. [shadowing and specification, smooth and hyperbolic dynamics][HIGH] Simultaneous renormalization of commuting circle diffeomorphismshttps://dyntopo.pages.dev/#2026-10-06/2610.050492026-10-09T08:26:07.415775ZBoris Petkovi\'cA renormalization operator for pairs of commuting circle diffeomorphisms, driven by the Brun continued fraction algorithm, converges to rigid rotations. This gives simultaneous smooth conjugacy to rotations for a full-measure class of rotation vectors. [circle and one-dimensional dynamics, smooth and hyperbolic dynamics][HIGH] Unbounded orbits in $C^1$-generic centrally symmetric strictly convex outer billiardshttps://dyntopo.pages.dev/#2026-10-06/2610.064182026-10-09T08:26:07.415775ZAlfonso SorrentinoFor a residual set of centrally symmetric strictly convex \(C^1\) tables, outer billiard maps have orbits escaping to infinity. This answers the Moser–Neumann question in that setting. [planar and surface dynamics, genericity][Related] Entropic rates from eigenfunction structure for $\times 2 \times 3$https://dyntopo.pages.dev/#2026-10-06/2610.047452026-10-09T08:26:07.415775ZPeter Burton, Kate JuschenkoAn effective Rudolph–Johnson bound is read as a constraint on eigenfunctions in potential counterexamples to Furstenberg's times-2 times-3 conjecture. [ergodic theory, entropy and chaos, circle and one-dimensional dynamics][Related] On global Lyapunov functions being Morsehttps://dyntopo.pages.dev/#2026-10-06/2610.056262026-10-09T08:26:07.415775ZWouter JongeneelFor a globally asymptotically stable hyperbolic equilibrium a Morse Lyapunov function exists, but not for every globally asymptotically stable equilibrium. [flows and foliations][Related] Temporal Metric Spaces and Temporal Iterated Function Systemshttps://dyntopo.pages.dev/#2026-10-06/2610.058052026-10-09T08:26:07.415775ZAmal P. S., Vinod Kumar P. B., Ramkumar P. BThe authors define interval-dependent metrics on trajectory spaces of evolving domains, study their topology and hyperspaces, and develop iterated function systems on them. [hyperspaces and descriptive topology, fractals and dimension][Related] On the centralizer and normalizer groups of odometers and Toeplitz subshiftshttps://dyntopo.pages.dev/#2026-10-06/2610.064882026-10-09T08:26:07.415775ZJaime G\'omez, Samuel PetiteFor residually finite groups, the authors study automorphism and normalizer groups of odometers and Toeplitz subshifts, with an orbit-equivalence result for \(\mathbb Z^d\) and flexible examples for Toeplitz subshifts. [symbolic dynamics, group actions][Related] The Almost Finite--Purely Infinite Dichotomy\\ for Minimal Amenable Ample Groupoidshttps://dyntopo.pages.dev/#2026-10-06/2610.067132026-10-09T08:26:07.415775ZG\'abor Elek, \'Ad\'am Tim\'arA minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite exactly when it has an invariant probability measure, and purely infinite otherwise. [group actions, minimal systems and factors][Related] Classifiability of crossed productshttps://dyntopo.pages.dev/#2026-10-06/2610.065862026-10-09T08:26:07.415775ZEusebio GardellaA survey of crossed products of topological dynamical systems and their classifiability in the Elliott program, aimed both as an entry point and a reference. [survey, group actions, minimal systems and factors][Related] Bratteli diagrams, translation flows and their $C^*$-algebrashttps://dyntopo.pages.dev/#2026-10-06/2205.015372026-10-09T08:26:07.415775ZIan F. Putnam, Rodrigo Trevi\~noThe authors connect the path space of a bi-infinite Bratteli diagram with the translation surface built from it, and relate the associated C*-algebras and their K-theory. [symbolic dynamics, flows and foliations][HIGH] Open dynamical systems in topologically expansive decreasing Lorenz mapshttps://dyntopo.pages.dev/#2026-10-05/2610.033272026-10-09T08:24:51.441170ZYiming Ding, Wolfgang Steiner, Yun SunThe authors develop an admissibility theory for topologically expansive decreasing Lorenz maps and use it to study the negative doubling map with a hole. The Hausdorff dimension of the survivor set is a devil's staircase in the right endpoint of the hole. [interval and graph maps, symbolic dynamics, fractals and dimension][HIGH] Sturmian beta-shifts do not have typical periodic optimizationhttps://dyntopo.pages.dev/#2026-10-05/2610.036192026-10-09T08:24:51.441170ZWen Huang, Oliver Jenkinson, Leiye Xu, Yiwei ZhangBeta-shifts whose lexicographically largest point is Sturmian do not have typical periodic optimization: an open set of Lipschitz functions is maximized only by the Sturmian measure. These are the first shifts with specification known to lack this property. [symbolic dynamics, shadowing and specification, genericity][Related] Near full groups of bounded type, IIhttps://dyntopo.pages.dev/#2026-10-05/2607.265722026-10-09T08:24:51.441170ZZheng KuangThe author gives criteria for when a group of bounded type contains the AF alternating group of the tail groupoid, and shows that this containment is not determined by the groupoid of germs alone. [group actions, minimal systems and factors]