This report covers 28 unseen announcements: 1 HIGH PRIORITY, 7 RELATED / POSSIBLY INTERESTING, and 20 LOW PRIORITY. It was reconstructed from an archived math.DS listing, so math.GN-only papers from this day are not included. The high-priority paper shows that expansive measures exist for the shift on the Hilbert cube and are generic in a dense G-delta sense. Related papers treat regional proximality for commutative semigroup actions and its combinatorial use, intermediate entropies for amenable actions, non-rigid disk pseudo-rotations, piecewise-linear circle actions, and two celestial-mechanics results. Claims are restricted to the supplied abstracts.
This day is part of week 2026-W41. Only new and newly cross-listed math.DS and math.GN papers are included. Papers by followed authors receive HIGH PRIORITY. Replacement-only submissions and previously reported arXiv IDs are excluded. Summaries are based on the title and abstract; the “In the paper” sections of high-priority papers and top picks also draw on the paper itself.
Expansiveness, topological stability, the Hilbert cube and a dense G-delta genericity statement combine several core topological themes.
TL;DR
The 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.
Problem
Decide whether Mañé's dimensional constraint for expansive homeomorphisms extends to measure-theoretic expansiveness, a question of Morales and Sirvent.
Main result
The shift map on the Hilbert cube is expansive with respect to the product Lebesgue measure, answering the question negatively. Any homeomorphism admitting a full-support expansive measure is densely measure-expansive. For the shift, the set of expansive and topologically stable measures contains a dense subset.
Methods / framework
Not specified in the abstract.
Context
Expansiveness, measure expansiveness, topological stability and infinite-dimensional compacta.
A classical theorem by Mañé states that compact metric spaces admitting expansive homeomorphisms must be finite-dimensional. It is a natural question, raised by Morales and Sirvent, whether this dimensional constraint extends to the measure-theoretic notion of expansiveness. In this paper, we answer this question in the negative. We prove that the shift map on the Hilbert cube is expansive with respect to the product Lebesgue measure. Furthermore, we establish a general sufficient condition: any homeomorphism admitting a full-support expansive measure is densely measure-expansive. As a consequence, we show that the set of expansive and topologically stable measures for the shift map contains a dense subset, revealing a rich measure-theoretic structure despite the infinite dimensionality of the phase space.
entropy and chaosgroup actionsshadowing and specification
Relevance
Entropy realization via an orbit-tracing (product-type) property for amenable group actions is related to specification-type properties.
TL;DR
An approximate product property for amenable group actions yields entropy-denseness, and with asymptotic entropy expansiveness every entropy in is realized by a residual set of ergodic measures.
Problem
Realize prescribed entropies by ergodic measures for continuous actions of infinite countable amenable groups.
Main result
The introduced approximate product property implies entropy-denseness and almost entropy-approximability of every invariant measure. Under asymptotic entropy expansiveness this upgrades to entropy-approximability, and for every ergodic measures of entropy form a residual subset of the invariant measures with entropy at least ; the set of ergodic measure entropies equals .
Methods / framework
Zero-entropy exact tilings. · A finite-block estimate for the complexity of tracing mistakes.
Context
Entropy theory, specification-type properties and amenable group actions.
We study entropy realization for continuous actions of infinite countable amenable groups. We introduce an approximate product property for amenable group actions that permits a small proportion of tracing mistakes on prescribed pairwise disjoint, sufficiently invariant finite sets. We prove that this property implies entropy-denseness and almost entropy-approximability of every invariant measure. The construction combines zero-entropy exact tilings with a finite-block estimate for the complexity of tracing mistakes. Under asymptotic entropy expansiveness, almost entropy-approximability upgrades to entropy-approximability. For every , ergodic measures of entropy then form a residual subset of the invariant measures whose entropy is at least . In particular, the set of ergodic measure entropies equals .37A35, 37B40, 37B05
Celestial mechanics; a universality result for Poincaré maps of restricted many-body problems.
TL;DR
Every symplectic embedding of the disk into the plane is approximated by a renormalized Poincaré map of a suitable restricted planar circular many-body problem.
Problem
Determine whether universal maps exist in celestial mechanics.
Main result
Every symplectic embedding of the disk into can be approximated with arbitrary precision by a renormalization of a Poincaré map of a restricted planar circular -body problem, for suitable and masses of the primaries. The authors call this a weak, finite-dimensional form of universal dynamics.
Methods / framework
Gonchenko–Shilnikov–Turaev theory. · Control of dynamics near homoclinic tangencies of arbitrarily high order.
Context
Celestial mechanics, universal dynamics and homoclinic tangencies.
Universal maps (maps whose renormalized iterations approximate every map in a given class) are locally generic in several spaces of diffeomorphisms [Bonatti–Díaz 2003, Turaev 2015]. In fluid dynamics, steady Euler flows whose Poincaré maps are universal are also known to be locally dense [Berger–Florio–PeraltaSalas, 2023]. Motivated by Arnold's vision [Arnold, 1966] that the complexity of orbits in celestial mechanics and that in fluids should be comparable, in the present paper we address the question of the existence of universal maps in celestial mechanics. We give a partial answer by proving a weak, finite-dimensional form of universal dynamics. More concretely, we show that every symplectic embedding of the disk into can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular -body problem, for suitable and appropriate choice of the masses of the primaries. The proof relies on Gonchenko–Shilnikov–Turaev theory [Turaev 2003, Gonchenko–Turaev–Shilnikov 2007] and requires control of the dynamics near homoclinic tangencies of arbitrarily high order; the techniques developed in [Garrido–Martín–Paradela, 2025] are essential for such control.
Constructions of area-preserving disk homeomorphisms that are not C^0-rigid relate to planar dynamics and approximation-by-conjugation constructions.
TL;DR
The authors construct area-preserving irrational pseudo-rotations of the closed disk that are not -rigid, arbitrarily close to the rigid rotation in or .
Problem
Construct non-rigid disk pseudo-rotations with controlled regularity and rotation number.
Main result
For every Brjuno rotation number there are area-preserving irrational pseudo-rotations of the closed disk that are not -rigid, arbitrarily -close to the rigid rotation. For , a weighted Brjuno condition gives examples, covering every rotation number with denominator-growth exponent below and some at the critical exponent. For a suitable Brjuno–Liouville rotation number there is an example whose derivative growth is not bounded by any polynomial.
Methods / framework
Not specified in the abstract.
Context
Disk pseudo-rotations, rigidity and arithmetic conditions on rotation numbers.
Subjects
math.DS
Keywords
pseudo-rotations · disk homeomorphisms · rigidity · Brjuno condition
We construct area-preserving irrational pseudo-rotations of the closed disk that fail to be -rigid. For every Brjuno rotation number we obtain examples, and they may be chosen arbitrarily -close to the rigid rotation with the same rotation number. For , a weighted Brjuno condition gives examples that are arbitrarily close in the topology; this weighted range contains every rotation number with denominator-growth exponent strictly smaller than and also some numbers at the critical exponent. For a suitable Brjuno–Liouville rotation number, we also obtain an example whose derivative growth is not bounded by any polynomial.
Celestial mechanics: variational construction of periodic and quasi-periodic N-body orbits using symmetry.
TL;DR
Rotational symmetry yields families of periodic and quasi-periodic N-body trajectories via variational persistence results.
Problem
Construct periodic and quasi-periodic orbits in rotationally symmetric N-body problems.
Main result
In the strong-force setting, Montgomery's variational approach is extended to free homotopy classes of curves periodic up to a prescribed nontrivial rotation. In the weak-force setting, under suitable hypotheses, global and local variational persistence holds for action minimizers in loop spaces defined by finite symmetry-group actions under sufficiently small rotations. Applications include the figure-eight orbit (under a numerically supported strict local minimality assumption) and double choreographic loops.
Methods / framework
Montgomery's variational approach. · Action minimization in symmetric loop spaces.
Context
Celestial mechanics, variational methods and choreographies.
Subjects
math.DS
Keywords
N-body problem · variational methods · choreographies · symmetry
Many mechanical systems, such as the Newtonian -body system, are invariant under rotations. The rotational symmetry of the system gives rise to two commuting actions: the dynamical action and the action of the symmetry group. In this paper, we exploit this symmetry to construct families of periodic and quasi-periodic trajectories. In the strong-force setting, we extend Montgomery's variational approach to free homotopy classes of curves that are periodic up to a prescribed nontrivial rotation. In the weak-force setting, we establish, under suitable hypotheses, global and local variational persistence results for action minimizers in loop spaces defined by general finite symmetry-group actions under sufficiently small rotations. As applications, we apply these results to the figure-eight orbit, under a numerically supported assumption of strict local minimality, to double choreographic loops with -fold rotation symmetry, and to counter-rotating double choreographic loops in planar -body problems.
Angelina Blahodatna, Lauren Detmold, Daniel Glasscock, Anh N. Le
minimal systems and factorsgroup actions
Relevance
Regional proximality and maximal equicontinuous factors of minimal systems are classical topological dynamics.
TL;DR
For minimal actions of commutative semigroups, the regionally proximal and equicontinuous structure relations are equivalence relations and coincide.
Problem
Extend the coincidence of the regionally proximal and equicontinuous structure relations from minimal abelian group actions to commutative semigroup actions.
Main result
For minimal actions of commutative semigroups both relations are equivalence relations and coincide. Using natural extensions for actions by surjections, the maximal equicontinuous factor of a minimal commutative semigroup action equals that of the group action into which it embeds. The authors report formal verification of all results in Lean.
Methods / framework
Natural extensions for commutative semigroup actions by surjections. · Formal verification in Lean, as reported by the authors.
Context
Minimal systems, proximality and equicontinuous factors.
The regionally proximal and equicontinuous structure relations are fundamental relations in topological dynamics that capture the equicontinuous behavior in a topological dynamical system and its factors. For minimal actions of abelian groups, these relations are known to be equivalence relations and are known to coincide. In this paper, we generalize these facts to semigroup actions: for minimal actions of commutative semigroups, the regionally proximal and equicontinuous structure relations are equivalence relations and the two coincide. We also develop the machinery of natural extensions for commutative semigroup actions that act by surjections, concluding that the maximal equicontinuous factor of a minimal action of a commutative semigroup is the same as the maximal equicontinuous factor of the group action into which it embeds. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.
Angelina Blahodatna, Lauren Detmold, Daniel Glasscock, Anh N. Le
minimal systems and factors
Relevance
Applies regional proximality for minimal commutative semigroup actions to the combinatorics of difference sets; topological dynamics as a tool.
TL;DR
Sets meeting every difference subset of a commutative semigroup have local Bohr structure.
Problem
Extend results linking difference sets and group rotations from the integers to arbitrary commutative semigroups.
Main result
Sets with non-empty intersection with all difference subsets of a commutative semigroup possess local Bohr structure, generalizing results of Bergelson–Furstenberg–Weiss and Host–Kra. The precise notion of local Bohr structure is not specified in the abstract. The authors report Lean verification.
Methods / framework
Regional proximality as an equivalence relation for minimal commutative semigroup actions. · A DeMorgan-type algebra on Furstenberg families.
Context
Topological dynamics, Furstenberg families and combinatorics of difference sets.
In this paper, we strengthen the connection between the combinatorics of difference sets and the dynamics of group rotations. Our main result shows that sets which have non-empty intersection with all difference subsets of a commutative semigroup possess local Bohr structure. This generalizes results of Bergelson, Furstenberg, and Weiss and Host and Kra from the integers to arbitrary commutative semigroups. We accomplish this by A) utilizing a recent result showing that the regionally proximal relation is an equivalence relation for minimal actions of commutative semigroups and by B) describing a new, DeMorgan-type algebra on Furstenberg families that allows for efficient manipulation and computation. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.
DSarXiv:2608.12168 · replace-cross · medium confidence
Leonardo Dinamarca, Maximiliano Escayola, Sang-hyun Kim, Thomas Koberda
group actionscircle and one-dimensional dynamics
Relevance
Structural restrictions on groups acting by piecewise-linear homeomorphisms of the circle; group actions on one-dimensional spaces.
TL;DR
Finitely generated groups of PL circle homeomorphisms satisfy a trichotomy, giving a free-or-surface alternative for word-hyperbolic such groups.
Problem
Identify group-theoretic constraints on piecewise-linear actions on the circle.
Main result
Every finitely generated subgroup of the ambient PL group (its notation is not defined in the abstract) is virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or one-ended, non-acylindrically-hyperbolic and containing . In particular word-hyperbolic groups of PL circle homeomorphisms satisfy a free-or-surface alternative.
Methods / framework
Not specified in the abstract.
Context
Group actions on the circle and hyperbolic groups.
Subjects
math.GR · math.DS
Keywords
circle actions · PL homeomorphisms · hyperbolic groups
We prove the following “Torus Trichotomy Theorem”: every finitely generated subgroup of is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing . In particular, we obtain the “free or surface” alternative for word-hyperbolic groups of PL homeomorphisms of the circle. [Source notation: unrecognized commands pls; their literal command names are retained; no definitions are inferred.]
subset.
is realized by a residual set of ergodic measures.
ergodic measures of entropy 
form a residual subset of the invariant measures with entropy at least 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)
.
, ergodic measures of entropy 
can be approximated with arbitrary precision by a renormalization of a Poincaré map of a restricted planar circular 
-body problem, for suitable 
and masses of the primaries. The authors call this a weak, finite-dimensional form of universal dynamics.
can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular 
and 
-rigid, arbitrarily close to the rigid rotation in 
, a weighted Brjuno condition gives 
and some at the critical exponent. For a suitable Brjuno–Liouville rotation number there is an example whose derivative growth is not bounded by any polynomial.
-Body Problems
-fold rotation symmetry, and to counter-rotating double choreographic loops in planar 
-body problems.
sets via a new Furstenberg family algebra
. In particular word-hyperbolic groups of PL circle homeomorphisms satisfy a free-or-surface alternative.
is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing 
. In particular, we obtain the “free or surface” alternative for word-hyperbolic groups of PL homeomorphisms of the circle. [Source notation: unrecognized commands pls; their literal command names are retained; no definitions are inferred.]
having non-constant periodic orbits