This report covers 25 unseen announcements: 3 HIGH PRIORITY, 4 RELATED / POSSIBLY INTERESTING, and 18 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 rigidity of interval maps from their periodic orbits, the equivalence of shadowing and topological stability for linear dynamics, and classification complexity for homeomorphism groups of the interval, the Cantor space and the Hilbert cube. Related papers concern mean dimension versus entropy of open covers, entropy and multiplicative endomorphisms of the circle, mixing of the BCZ map, and comparison for ample groupoids. 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.
Rigidity of topologically mixing interval maps and of basic sets in the spectral decomposition is core one-dimensional topological dynamics.
TL;DR
Periodic 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.
Problem
Determine to what extent periodic orbits determine a one-dimensional dynamical system.
Main result
Periodic orbits determine every topologically mixing interval map. For a non-mixing interval map with dense periodic points, the periodic orbits need not determine but always determine . For maps on compact intervals, for each basic set in the spectral decomposition of there is an iterate whose restriction to is determined by the periodic orbits of .
Methods / framework
Not specified in the abstract.
Context
Interval maps, periodic orbits, topological mixing and the spectral decomposition.
We study the extent to which periodic orbits determine a one-dimensional dynamical system. In particular, we show that they determine every topologically mixing interval map. More generally, for a non-mixing interval map with dense periodic points, the periodic orbits need not determine , but they always do so for . Finally, for maps on compact intervals, we obtain a local version of this rigidity: for each basic set arising in the spectral decomposition of an interval map , there exists an iterate whose restriction to is determined by the periodic orbits of .
Shadowing and topological stability are central notions in the profile; the paper characterizes both for linear dynamics on Banach spaces.
TL;DR
For 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.
Problem
Characterize the shadowing property of invertible bounded linear operators, for which generalized hyperbolicity is too restrictive.
Main result
The authors introduce generalized pseudo-hyperbolicity, formulated through continuous homogeneous maps generating exponentially decaying Green families. For arbitrary sequences of invertible bounded linear operators on Banach spaces, it is equivalent to both shadowing and topological stability. In the autonomous setting this gives the equivalence of shadowing and topological stability without the additional assumptions of earlier results. Shadowing and topological stability are robust for such sequences.
Methods / framework
Continuous homogeneous maps generating exponentially decaying Green families.
Context
Shadowing, topological stability, and autonomous and nonautonomous linear dynamics.
Although generalized hyperbolicity has proved to be a fundamental notion in linear dynamics, recent results show that it is too restrictive to characterize the shadowing property of invertible bounded linear operators on Banach spaces. This motivates the introduction of generalized pseudo-hyperbolicity, a weaker notion formulated through continuous homogeneous maps generating exponentially decaying Green families. We prove that, for arbitrary sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to both shadowing and topological stability. In the autonomous setting, this yields the equivalence between shadowing and topological stability, removing additional assumptions from previous results. We also prove robustness of shadowing and topological stability for sequences of invertible bounded linear operators.
DSarXiv:2608.30346 · replace-cross · medium confidence
Homeomorphism groups of compacta, hyperspace actions and the Hilbert cube connect continuum theory and topological dynamics with descriptive set theory.
TL;DR
Replacing 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.
Problem
Understand how the classification complexity of orbit equivalence relations changes when is replaced by a dense non-closed subgroup.
Main result
For a compact space and , the authors study the left shift action on , the hyperspace action on and the conjugation action on . For bi-Lipschitz homeomorphisms, diffeomorphisms and bi-absolutely continuous homeomorphisms inside , passing to the subgroup strictly increases the complexity of the classification problems or makes them incomparable with the full-group relations, in contrast to closed subgroups. A similar behaviour occurs for hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure.
Methods / framework
Not specified in the abstract.
Context
Descriptive set theory, homeomorphism groups, hyperspaces and orbit equivalence relations.
Subjects
math.LO · math.DS
Keywords
homeomorphism groups · hyperspaces · classification complexity · Hilbert cube · Cantor space
In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space and a subgroup , we consider three canonical actions: the left shift action on , the induced hyperspace action on , and the conjugation action on We first analyze subgroups of the group of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.
ergodic theoryentropy and chaoscircle and one-dimensional dynamics
Relevance
Entropy and limits of measures under multiplicative endomorphisms of the circle; one-dimensional ergodic theory around Furstenberg's conjecture.
TL;DR
Entropy alone, without invariance, pushes a measure on the circle toward Lebesgue measure under any sublacunary set of maps .
Problem
Prove a form of the Rudolph–Johnson theorem without an invariance hypothesis.
Main result
For every sublacunary set of multipliers (consecutive ratios tending to 1), every Borel probability measure on the circle has a weak-star limit of endomorphs dominating Lebesgue measure scaled by the upper entropy dimension of . The results are effective at finite entropy resolution, with bounds for the primes and perfect powers. An analogue of Lyons' conjecture holds at full entropy dimension for every sublacunary set and fails by an arbitrarily large factor for the powers of 3.
Methods / framework
Fourier analysis combined with the algebraic structure of multiplicative endomorphisms.
Context
Furstenberg's times-2 times-3 problem, entropy and equidistribution on the circle.
We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms . Here, we refer to an infinite set of multipliers as sublacunary if its consecutive ratios tend to . Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers and the semigroup generated by and which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of . These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of . The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.
Minimal ample groupoids generalize minimal Cantor actions; the focus is comparison properties motivated by C*-algebras.
TL;DR
Fiberwise 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.
Problem
Find conditions giving groupoid strict comparison for minimal ample groupoids.
Main result
For a minimal ample groupoid with compact unit space that is fiberwise supramenable or topologically amenable, the clopen type semigroup is almost unperforated; if is -compact it has groupoid strict comparison. Applications include Matui's AH conjecture and pure infiniteness.
Methods / framework
Not specified in the abstract.
Context
Ample groupoids, type semigroups and minimal Cantor systems.
Subjects
math.DS · math.OA
Keywords
ample groupoids · strict comparison · type semigroup · supramenability
In this paper, we first introduce fiberwise supramenability for locally compact Hausdorff étale groupoids with compact unit space. Then, for such a minimal ample groupoid , we show that if is fiberwise supramenable or topologically amenable, then the clopen type semigroup is almost unperforated. Consequently, if is -compact, then it has groupoid strict comparison. We then present several applications, including Matui's AH conjecture and pure infiniteness for groupoids and groupoid -algebras.
Entropy pairs, uniformly positive entropy and mean dimension of open covers are topological-dynamics invariants.
TL;DR
The 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.
Problem
Find a cover-level counterpart of the Lindenstrauss–Weiss comparison between entropy and mean dimension.
Main result
The mean dimension of a finite open cover is quantitatively bounded by its topological entropy and cardinality; a zero-entropy cover has zero mean dimension. The comparison holds for actions of countable amenable groups and in the sofic setting. Consequently every mean dimension pair is an entropy pair and UPMD implies UPE, answering Questions 2.27 and 4.9 of García-Ramos and Gutman.
Methods / framework
A polytope associated with a minimal subcover. · The Lindenstrauss–Weiss compression argument. · Maurey's empirical method for volumes of coordinate projections.
Context
Topological entropy, mean dimension, entropy pairs and local entropy theory.
Subjects
math.DS
Keywords
mean dimension · entropy pairs · open covers · uniformly positive entropy
We establish a counterpart, for individual finite open covers, of the global entropy-mean-dimension comparison of Lindenstrauss and Weiss. More precisely, the mean dimension of a finite open cover is quantitatively bounded in terms of its topological entropy and cardinality. In particular, a finite open cover with zero entropy has zero mean dimension. The same comparison holds for actions of countable amenable groups and in the sofic setting. It follows that every mean dimension pair is an entropy pair and, in particular, that uniformly positive mean dimension (UPMD) implies uniformly positive entropy (UPE). The cover and pair implications answer Questions 2.27 and 4.9 of García-Ramos and Gutman, respectively. The proof associates a polytope to a minimal subcover and combines the Lindenstrauss-Weiss compression argument with Maurey's empirical method to estimate the volumes of its coordinate projections.
with dense periodic points, the periodic orbits need not determine 
. For maps on compact intervals, for each basic set 
in the spectral decomposition of 
whose restriction to 
is replaced by a dense non-closed subgroup.
and 
, the authors study the left shift action on 
and the conjugation action on 
. For bi-Lipschitz homeomorphisms, diffeomorphisms and bi-absolutely continuous homeomorphisms inside ![\mathcal H^+([0,1])](data:image/svg+xml;base64,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)
, passing to the subgroup strictly increases the complexity of the classification problems or makes them incomparable with the full-group relations, in contrast to closed subgroups. A similar behaviour occurs for hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure.
, we consider three canonical actions: the left shift action on 
, the induced hyperspace action on 
, and the conjugation action on ![\mathcal{H}^+([0,1])](data:image/svg+xml;base64,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)
of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.
.
on the circle has a weak-star limit of endomorphs dominating Lebesgue measure scaled by the upper entropy dimension of 
. Here, we refer to an infinite set of multipliers 
as sublacunary if its consecutive ratios tend to 
. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers 
and the semigroup generated by 
and 
which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure 
with compact unit space that is fiberwise supramenable or topologically amenable, the clopen type semigroup 
is almost unperforated; if 
-compact it has groupoid strict comparison. Applications include Matui's AH conjecture and pure infiniteness.
, we show that if 
is almost unperforated. Consequently, if 
-algebras.
iterated function systems