This report covers 54 unseen announcements: 3 HIGH PRIORITY, 7 RELATED / POSSIBLY INTERESTING, and 44 LOW PRIORITY. It was reconstructed from an archived math.DS listing, so math.GN-only papers from this day are not included. High-priority themes are structural stability and shadowing beyond hyperbolicity, renormalization of commuting circle diffeomorphisms, and C^1-generic unbounded orbits of planar outer billiards. Related papers concern odometers and Toeplitz subshifts, minimal ample groupoids on the Cantor set, a survey of crossed products of topological dynamical systems, topology of Lyapunov functions, hyperspaces of trajectory spaces, times-2 times-3 entropy rates, and Bratteli diagrams for translation surfaces. 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.
shadowing and specificationsmooth and hyperbolic dynamics
Relevance
Shadowing and structural stability without expansivity are central notions in topological dynamics, here in a Banach-space setting.
TL;DR
A 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.
Problem
Establish stability results for Banach-space dynamics whose generalized-hyperbolic splittings may be discontinuous.
Main result
For generalized-hyperbolic diffeomorphisms in the uniform global category, sufficiently small bounded Lipschitz perturbations give continuous semiconjugacies in both directions; surjectivity is not asserted. For an infinite product of one-dimensional Morse–Smale systems, the authors obtain full structural stability without expansivity on every , , including perturbations coupling coordinates; these products have no compact global attractor and, for , uncountably many hyperbolic fixed points. The cocycle analysis gives Lipschitz shadowing for the induced operator on bounded continuous vector fields.
Methods / framework
Michael's continuous selection theorem. · Conversion of uniform orbitwise solvability estimates into bounded continuous solutions of cohomological equations. · Analysis of generalized-hyperbolic cocycles.
Context
Structural stability, generalized hyperbolicity, shadowing, and infinite-dimensional dynamics.
We introduce a new method based on Michael's continuous selection theorem for stability of Banach-space dynamics with possibly discontinuous generalized-hyperbolic splittings. The method converts uniform orbitwise solvability estimates into bounded continuous solutions of cohomological equations. For generalized-hyperbolic diffeomorphisms in the uniform global category, it yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations; surjectivity is not asserted. For an infinite product of one-dimensional Morse-Smale systems, we construct compatible forward and reverse cohomological solution operators and obtain full structural stability without expansivity. This result holds on every , , including perturbations that couple coordinates. These products have no compact global attractor and, for , have uncountably many hyperbolic fixed points. The continuous-selection argument underlying the general semiconjugacy theorem is developed for generalized-hyperbolic cocycles, yielding bounded continuous solvability of cohomological equations. The cocycle analysis also gives Lipschitz shadowing for the induced operator on bounded continuous vector fields.
circle and one-dimensional dynamicssmooth and hyperbolic dynamics
Relevance
Renormalization and rigidity for commuting circle diffeomorphisms is one-dimensional dynamics; the invariant lamination by topological conjugacy classes is a topological statement.
TL;DR
A 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.
Problem
Develop renormalization and rigidity theory for pairs of commuting orientation-preserving circle diffeomorphisms.
Main result
For rotation vectors of simultaneously bounded type, under a summable strong convergence condition (satisfied by all eventually periodic types with Pisot period matrix), renormalizations converge exponentially fast in to pairs of rigid rotations, and the pair is simultaneously conjugate to the rotations with depending only on the bounded type constant. A fast renormalization extends exponential convergence and rigidity to a full-measure class of rotation vectors. The operator is topologically hyperbolic, and the partition of renormalized pairs by rotation vector is an invariant lamination whose leaves are topological conjugacy classes. Results hold in class .
Methods / framework
Renormalization driven by the accelerated Brun multidimensional continued fraction algorithm. · A fast renormalization for rotation vectors outside bounded type.
Context
Circle diffeomorphisms, renormalization, rigidity and multidimensional continued fractions.
We define a renormalization operator for pairs of commuting orientation preserving circle diffeomorphisms of class . The operator is driven by the accelerated Brun multidimensional continued fraction algorithm. For rotation vectors of simultaneously bounded type, under a summable strong convergence condition satisfied by all eventually periodic types with Pisot period matrix, the renormalizations converge exponentially fast in to pairs of rigid rotations. The pair is simultaneously conjugate to the pair of rotations, with depending only on the bounded type constant. A fast renormalization extends exponential convergence and rigidity to a class of rotation vectors of full Lebesgue measure, containing explicit vectors outside bounded type. We show that the operator is topologically hyperbolic. The partition of the renormalized pairs by rotation vector is an invariant lamination whose leaves are the topological conjugacy classes. All results hold for pairs of class .
A residual-set (Baire category) result for planar area-preserving twist dynamics, built on invariant curves and Birkhoff regions of instability, matches the genericity and planar-dynamics parts of the profile.
TL;DR
For a residual set of centrally symmetric strictly convex tables, outer billiard maps have orbits escaping to infinity. This answers the Moser–Neumann question in that setting.
Problem
Decide whether outer billiards about strictly convex planar bodies can have unbounded orbits (a question of Moser and Neumann).
Main result
Let be the space of support functions of centrally symmetric, strictly convex, compact planar bodies with boundary, with the topology. For a residual subset of , the outer billiard map admits unbounded orbits escaping to infinity; this residual set contains support functions arbitrarily -close to those of disks or ellipses. A rigidity theorem states that if the radius-of-curvature measure of is purely singular, every continuous invariant tangent graph consists entirely of periodic points.
Methods / framework
A rigidity theorem for continuous invariant tangent graphs when the radius-of-curvature measure is purely singular. · Genericity argument showing these periodic invariant graphs are absent. · Mather's diffusing mechanism in a Birkhoff region of instability.
Context
Outer billiards, area-preserving twist maps of the plane, invariant curves and Baire-generic properties.
Subjects
math.DS
Keywords
outer billiards · residual sets · unbounded orbits · Birkhoff region of instability · invariant curves
Let be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a curve, endowed with the topology. We prove that for a residual subset of the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily -close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.
ergodic theoryentropy and chaoscircle and one-dimensional dynamics
Relevance
Entropy constraints for times-2 times-3 invariant measures on the circle; ergodic theory with a one-dimensional component.
TL;DR
An effective Rudolph–Johnson bound is read as a constraint on eigenfunctions in potential counterexamples to Furstenberg's times-2 times-3 conjecture.
Problem
Constrain the behaviour of eigenfunctions in potential counterexamples to Furstenberg's conjecture.
Main result
A near return of to itself under the semigroup generated by 2 and 3 forces the resolution entropy to be small, and adherence of this coordinate function to eigenfunctions of forces it to decay. The rates are explicit in the rate of adherence and the arithmetic of the eigenvalues.
Methods / framework
A finite-resolution form of the Rudolph–Johnson theorem from a companion article.
Context
Measure rigidity and entropy for times-2 times-3 invariant measures on the circle.
Subjects
math.DS
Keywords
times 2 times 3 · Furstenberg conjecture · entropy · eigenfunctions
In a companion article [RJnodyn] the present authors proved a form of the Rudolph–Johnson theorem which is effective at finite resolution, bounding the low-frequency Fourier coefficients of a -invariant measure by its normalized entropy. Here we read that bound in the other direction, as a constraint on eigenfunction behavior in potential counterexamples to Furstenberg's conjecture. We show that a near return of the coordinate function to itself under the semigroup generated by and forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.
A topological question about Lyapunov functions for flows, linked to the 4-dimensional smooth Poincaré conjecture.
TL;DR
For a globally asymptotically stable hyperbolic equilibrium a Morse Lyapunov function exists, but not for every globally asymptotically stable equilibrium.
Problem
Determine whether a smooth Lyapunov function for a globally asymptotically stable (GAS) equilibrium on Euclidean space can always be chosen Morse.
Main result
If the GAS equilibrium is hyperbolic, a Morse Lyapunov function exists. If the statement held in general, the generalized smooth Poincaré conjecture in dimension 4 would follow; the author gives a GAS example without a Morse Lyapunov function, closing this route.
Methods / framework
Not specified in the abstract.
Context
Lyapunov functions, Morse theory and the topology of flows.
Lyapunov functions have been a cornerstone of dynamical systems theory ever since the early 1900s. Moreover and akin to Morse theory, Lyapunov functions have been the key in linking topology to dynamical systems theory. In this note we continue along these lines. Suppose that some vector field on Euclidean space has a unique equilibrium point that is globally asymptotically stable (GAS). Under these conditions, it is known that there is always a smooth Lyapunov function to certify stability. One may wonder if this function can always be chosen to be Morse. We show that if the equilibrium point is hyperbolic, this is indeed true. Moreover, we show that if this would be true in general, then the generalized 4-dimensional smooth Poincaré conjecture (SPC4) must be true. Even so, we provide an example that is GAS, but does not admit a Morse Lyapunov function, closing this route of proving SPC4.
hyperspaces and descriptive topologyfractals and dimension
Relevance
Cross-listed to math.GN; hyperspaces and attractors of iterated function systems on evolving metric spaces have a general-topology component.
TL;DR
The authors define interval-dependent metrics on trajectory spaces of evolving domains, study their topology and hyperspaces, and develop iterated function systems on them.
Problem
Formalize metric and topological structure on continuously evolving domains and extend iterated function systems to that setting.
Main result
The trajectory space inherits completeness from the underlying metric space, and the corresponding temporal hyperspace is complete. Attractors of the temporal iterated function systems evolve coherently with the underlying dynamics and remain stable even when the generating maps lose the classical contraction property.
Methods / framework
Not specified in the abstract.
Context
Metric spaces, hyperspaces and iterated function systems.
Subjects
math.DS · math.GN · math.MG
Keywords
hyperspaces · iterated function systems · trajectory spaces · completeness
We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.
Odometers and Toeplitz subshifts are standard symbolic models in topological dynamics; the paper studies their automorphism and normalizer groups.
TL;DR
For residually finite groups, the authors study automorphism and normalizer groups of odometers and Toeplitz subshifts, with an orbit-equivalence result for and flexible examples for Toeplitz subshifts.
Problem
Understand the automorphism (centralizer) and normalizer groups of odometers and Toeplitz subshifts over residually finite groups.
Main result
Two continuously orbit equivalent -odometers have commensurable normalizers when , but this fails for other nilpotent groups. The automorphism and normalizer groups of any Toeplitz subshift embed into those of its odometer, yet there are low-complexity -Toeplitz subshifts whose centralizer is isomorphic to (or restricted to its center) while the normalizer is as large as possible.
Methods / framework
Not specified in the abstract.
Context
Symbolic dynamics, odometers, Toeplitz subshifts and continuous orbit equivalence.
For residually finite groups, we study the automorphism and normalizer groups of odometers and their symbolic extensions: Toeplitz subshifts. In relation to orbit equivalence theory, we show that two continuous orbit equivalent -odometers have commensurable normalizers when , but that this result fails for other nilpotent groups . The automorphism and normalizer groups of any Toeplitz subshift embed into those of its underlying odometer; despite this constraint, they exhibit considerable flexibility. For instance, we exhibit examples of low-complexity -Toeplitz subshifts whose centralizer is isomorphic to (or, at the other extreme, restricted to its center) yet whose normalizer group is as large as possible.
Minimal amenable groupoids on the Cantor set and invariant measures are topological dynamics of group actions, though the motivation comes from operator algebras.
TL;DR
A 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.
Problem
Answer Matui's question on the almost finite–purely infinite dichotomy for minimal amenable ample groupoids, in the principal setting.
Main result
A second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; without such a measure it is purely infinite. Every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite.
Methods / framework
The three-to-two comparison method of Glasner and Liu. · Randomized Følner packing methods of Elek and Timár.
Context
Ample groupoids, minimal Cantor systems, comparison and almost finiteness.
Subjects
math.DS · math.OA
Keywords
ample groupoids · almost finiteness · Cantor set · invariant measures · comparison
We answer, in the principal setting, a question of Matui on the almost finite–purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.
An expository survey translating C*-algebraic classification into topological-dynamical conditions (minimality, freeness, amenability); surveys of topological dynamics are in the profile.
TL;DR
A survey of crossed products of topological dynamical systems and their classifiability in the Elliott program, aimed both as an entry point and a reference.
Problem
Survey when C*-crossed products of topological dynamical systems are classifiable, and which dynamical properties ensure this.
Main result
The survey presents the crossed product construction for discrete group actions and, for commutative systems, the facts that nuclearity corresponds to amenability of the action and simplicity to minimality plus topological freeness. It reviews dynamical tools for -stability for amenable and nonamenable groups, two conjectures expected to describe the full picture, and central open problems.
Methods / framework
Expository survey.
Context
Topological dynamics and C*-algebras; the Elliott classification program.
This survey studies C*-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality, and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang-Su algebra . We present the main dynamical tools known to obtain -stability, both in the settings of amenable and nonamenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.
Explicit topological links between Bratteli path spaces and translation surfaces; Bratteli diagrams are symbolic models of minimal Cantor systems.
TL;DR
The 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.
Problem
Relate the path space of a bi-infinite Bratteli diagram to the translation surface constructed from it.
Main result
Explicit links between the path space and the surface, through intermediate topological spaces, relate the C*-algebras of tail equivalence and of the surface foliation under mild hypotheses, and their K-theory. For finite genus, Rauzy–Veech induction and its inverse give an explicit construction of the Bratteli diagrams.
Methods / framework
Rauzy–Veech induction.
Context
Bratteli diagrams, translation flows and operator algebras.
In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the -algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the -algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.
category, sufficiently small bounded Lipschitz perturbations give continuous semiconjugacies in both directions; surjectivity is not asserted. For an infinite product of one-dimensional Morse–Smale systems, the authors obtain full structural stability without expansivity on every 
, 
, including perturbations coupling coordinates; these products have no compact global attractor and, for 
, uncountably many hyperbolic fixed points. The cocycle analysis gives Lipschitz shadowing for the induced operator on bounded continuous vector fields.
, 
, including perturbations that couple coordinates. These products have no compact global attractor and, for 
to pairs of rigid rotations, and the pair is simultaneously 
conjugate to the rotations with 
depending only on the bounded type constant. A fast renormalization extends exponential convergence and 
.
. The operator is driven by the accelerated Brun multidimensional continued fraction algorithm. For rotation vectors of simultaneously bounded type, under a summable strong convergence condition satisfied by all eventually periodic types with Pisot period matrix, the renormalizations converge exponentially fast in 
be the space of support functions of centrally symmetric, strictly convex, compact planar bodies with 
is purely singular, every continuous invariant tangent graph consists entirely of periodic points.
has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in 
conjecture.
to itself under the semigroup generated by 2 and 3 forces the resolution entropy to be small, and adherence of this coordinate function to eigenfunctions of 
forces it to decay. The rates are explicit in the rate of adherence and the arithmetic of the eigenvalues.
to itself under the semigroup generated by 
and 
forces the resolution entropy to be small, and that adherence of the coordinate function to eigenfunctions of 
forces it to decay. These rates are explicit in the rate of adherence and in the arithmetic of the eigenvalues.
and flexible examples for Toeplitz subshifts.
-odometers have commensurable normalizers when 
, but this fails for other nilpotent groups. The automorphism and normalizer groups of any Toeplitz subshift embed into those of its odometer, yet there are low-complexity 
, but that this result fails for other nilpotent groups 
-stability for amenable and nonamenable groups, two conjectures expected to describe the full picture, and central open problems.
. We present the main dynamical tools known to obtain 
-algebras
-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the 
Endpoint