This report covers 21 unseen announcements: 2 HIGH PRIORITY, 1 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. The central themes are symbolic dynamics of decreasing Lorenz maps with holes and the failure of typical periodic optimization on Sturmian beta-shifts with specification; a related paper concerns topological full groups and minimal actions. 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.
interval and graph mapssymbolic dynamicsfractals and dimension
Relevance
Kneading theory and survivor sets for expansive decreasing Lorenz maps are one-dimensional symbolic dynamics, central to the interval-map side of the profile.
TL;DR
The 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.
Problem
Describe admissible kneading data for topologically expansive decreasing Lorenz maps and the survivor sets of the negative doubling map with a hole .
Main result
Every decreasing-admissible pair is realized by embedding into the circle map . For with a hole , , plateaux of the symbolic survivor spaces are classified via the alternating lexicographic order, and under weak admissibility and strict shifted ordering the boundary subshifts are kneading spaces of decreasing Lorenz maps. For fixed , the map is a devil's staircase, possibly constant. Under the pullback-null condition (P), a corresponding survivor-entropy result holds for expansive decreasing Lorenz maps; the condition is verified for a piecewise-linear family.
Methods / framework
Embedding into the circle map . · Alternating lexicographic order on itineraries. · Entropy correspondences across countable itinerary discrepancies in completed survivor systems.
Context
Kneading theory of Lorenz-type interval maps, open dynamical systems with holes, and entropy and dimension of survivor sets.
Subjects
math.DS
Keywords
Lorenz maps · kneading theory · open dynamical systems · survivor sets · devil's staircase
We develop a parity-sensitive admissibility theory for topologically expansive decreasing Lorenz maps. By embedding them into the standard circle map , we realize every decreasing-admissible pair. We then study the negative doubling map with a hole , where . Using the alternating lexicographic order, we classify plateaux of the symbolic survivor spaces. Under weak admissibility and strict shifted ordering, the boundary subshifts are realized as kneading spaces of decreasing Lorenz maps. We also establish entropy correspondences across countable itinerary discrepancies in completed survivor systems. For fixed , we prove that is a devil's staircase, possibly constant. Under the pullback-null condition (P), we obtain the corresponding survivor-entropy result for expansive decreasing Lorenz maps. We verify this condition for the piecewise-linear family in Example 5.1.
Wen Huang, Oliver Jenkinson, Leiye Xu, Yiwei Zhang
symbolic dynamicsshadowing and specificationgenericity
Relevance
Beta-shifts with the specification property and a genericity statement about maximizing measures touch both the specification and the Baire-category parts of the profile.
TL;DR
Beta-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.
Problem
Decide whether beta-shifts have typical periodic optimization (TPO), i.e. whether Lipschitz functions with a periodic maximizing measure contain an open dense set.
Main result
If the lexicographically largest point of a beta-shift is a Sturmian sequence, there is a non-empty open set of Lipschitz functions whose unique maximizing measure is the Sturmian measure, so TPO fails. These are the first beta-shifts without TPO and, since they have specification, the first shift spaces with specification but without TPO. The corresponding beta-transformations lack TPO in every space of Hölder functions on the interval.
Methods / framework
A rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the Lipschitz functions on such a subshift form a one-dimensional space.
Context
Ergodic optimization, beta-transformations of the interval, Sturmian subshifts, and the specification property.
A shift space is said to have typical periodic optimization (TPO) if the set of Lipschitz functions whose unique maximizing measure is supported on a periodic orbit contains an open dense subset of the space of Lipschitz functions. We show that beta-shifts whose lexicographically largest point is a Sturmian sequence do not have TPO: on each such beta-shift there is a non-empty open set of Lipschitz functions, all of which have the Sturmian measure as their unique maximizing measure. These are the first known examples of beta-shifts without TPO, and, since they have the specification property, the first known examples of shift spaces with specification but without TPO. The corresponding beta-transformations do not have TPO in any space of Hölder functions on the interval. The main ingredient in the proof of these results is a rigidity property of Sturmian subshifts: modulo constants and Lipschitz coboundaries, the space of Lipschitz functions on such a subshift is one-dimensional.
Related / Possibly Interesting
DSarXiv:2607.26572 · replace-cross · medium confidence
Topological full groups, groupoids of germs and minimal actions belong to topological dynamics and group actions, though the focus is group-theoretic.
TL;DR
The 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.
Problem
Determine when the AF alternating group of the tail groupoid is contained in the relevant groups, the AF input needed in the author's full-group completion theorem.
Main result
A sufficient criterion is given, in which selectors separate inherited diagonal actions, together with an exact criterion for finite regular coverings; in the covering case the intersection with the AF alternating group is identified as a centralizer. There are finitely generated examples with minimal actions and identical groupoids of germs but different intersections with the AF alternating group. A fragmentation group of the modified Fabrykowski–Gupta group contains the AF alternating group and has index four in its topological full group.
Methods / framework
Selectors separating inherited diagonal actions. · Fragmentations producing the required selectors. · The author's full-group completion theorem.
Context
Topological full groups, groupoids of germs, and minimal group actions on Cantor-type spaces.
Subjects
math.GR · math.DS
Keywords
topological full groups · groupoids of germs · AF alternating group · minimal actions
Containment of the AF alternating group of the tail groupoid is the AF input in our full-group completion theorem for groups satisfying the finite singular germ condition. We give a sufficient criterion for this containment, in which selectors separate inherited diagonal actions, and an exact criterion for finite regular coverings. We then identify a class of fragmentations that produce the required selectors. In the covering case, the intersection with the AF alternating group is the centralizer of the subgroup of deck transformations commuting with the adjoined AF permutations. This yields finitely generated examples with minimal actions and identical groupoids of germs but different intersections with the AF alternating group. Consequently, containment of the AF alternating group is not determined by the groupoid of germs alone. As an application, we show that a fragmentation group of the modified Fabrykowski–Gupta group contains the AF alternating group and, by the completion theorem, has index four in its topological full group.
with a hole 
.
. For 
, plateaux of the symbolic survivor spaces are classified via the alternating lexicographic order, and under weak admissibility and strict shifted ordering the boundary subshifts are kneading spaces of decreasing Lorenz maps. For fixed 
, the map 
is a devil's staircase, possibly constant. Under the pullback-null condition (P), a corresponding survivor-entropy result holds for expansive decreasing Lorenz maps; the condition is verified for a piecewise-linear family.
. Using the alternating lexicographic order, we classify plateaux of the symbolic survivor spaces. Under weak admissibility and strict shifted ordering, the boundary subshifts are realized as kneading spaces of decreasing Lorenz maps. We also establish entropy correspondences across countable itinerary discrepancies in completed survivor systems. For fixed