This report covers 44 unseen announcements from math.DS and math.GN: 1 HIGH PRIORITY, 8 RELATED / POSSIBLY INTERESTING, and 35 LOW PRIORITY. The high-priority paper studies local connectivity of the filled Julia set bundle over the Mandelbrot set. Related papers concern Hausdorff-continuity of Julia sets in non-archimedean and hybrid families, quantitative recurrence for expanding Markov interval maps including tent maps, variational principles for neutralized entropy, Conley indices of gradient flows, foliations forcing closed orbits, invariant sets and Hofer rigidity of surface diffeomorphisms, pseudo-Anosov flows, and a universal quasi-Polish space with its hyperspace. 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.
Local connectedness of a fibred planar compactum, with puzzle constructions and an explicit failure, is a continuum-theoretic question inside complex dynamics.
TL;DR
The 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.
Problem
Study the topology of , the model space for cubic capture straightening in the Inou–Kiwi framework.
Main result
The boundary of the non-escaping locus has Hausdorff dimension 4. The locus is locally connected at every with hyperbolic . For non-degenerate pairs with , shrinking of both planar puzzles implies local connectedness at . Local connectedness fails at some with real parabolic and , although both the parameter space and the filled Julia fibre are locally connected there.
Methods / framework
Mixed puzzle pieces built from Yoccoz puzzles and parapuzzles.
Context
Quadratic polynomials, the Mandelbrot set, filled Julia sets and local connectivity.
Subjects
math.DS
Keywords
local connectedness · filled Julia sets · Mandelbrot set · Yoccoz puzzles · Hausdorff dimension
In the paper extended reading of the paper
The author studies , the non-escaping locus of the skew product , and its restriction over the Mandelbrot set. Theorem 1.1: is closed, connected and completely invariant, the projection to the parameter is proper, and is a continuum. Theorem 1.2: is not locally connected at any with , and is locally connected at every with hyperbolic (via a local product structure from holomorphic motions).
Theorem 1.3 concerns non-degenerate pairs : outside the main cardioid and neither the critical orbit nor the orbit of hitting the dividing fixed point . The author builds mixed puzzle pieces by moving Yoccoz puzzle pieces holomorphically over Roesch's parapuzzle pieces (Słodkowski extension), and proves that the mixed pieces shrink to the point exactly when both the parapuzzle and the dynamical puzzle shrink; in that case is locally connected at . This covers non-renormalizable parameters with recurrent critical point and .
Theorem 1.4 answers the natural converse negatively: at the period-three primitive root there is where is not locally connected, although and are both locally connected there. The proof uses Lavaurs limits from parabolic implosion together with a "gate" lemma: real parabolic roots separate the real slice of , and nearby fibres stay a definite distance from . Theorem 1.5 gives from Shishikura's results and a Fubini-type slicing inequality of Héra–Keleti–Máthé. The motivation is the Inou–Kiwi capture straightening map, whose model space is ; the author notes the use of AI tools in preparing the manuscript.
For the quadratic family , we study the non-escaping locus , the model space for cubic capture straightening in the framework of Inou and Kiwi. Here is the Mandelbrot set, is the filled Julia set of , and . We prove that has full Hausdorff dimension . The set is locally connected at every with hyperbolic . At parameters on , we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of at . However, local connectedness fails at some with real parabolic and , even though both the parameter space and the filled Julia fiber are locally connected there. [Source notation: unrecognized commands KM, M; their literal command names are retained; no definitions are inferred.]
Identifies a Conley index (a topological invariant of isolated invariant sets of flows) with a suspension of the Milnor fibre.
TL;DR
For a holomorphic germ with an isolated critical point, the origin is an isolated invariant set of the negative gradient flow of , with Conley index the suspension of the Milnor fibre.
Problem
Compute the Conley index of the origin for the negative gradient flow of the real part of a holomorphic germ with an isolated critical point.
Main result
The origin is an isolated invariant set of the flow , and its Conley index is homotopy equivalent to the suspension of the Milnor fibre, a wedge of spheres . Hence the Milnor number equals the rank of the homological Conley index in degree , and the Poincaré–Hopf index of is recovered.
Methods / framework
Continuation to a holomorphic morsification. · The imaginary part of as a first integral, ruling out connecting orbits after a rotation. · The sum property of the Conley index and McCord's theorem.
Context
Conley index theory, gradient flows and singularity theory.
Let be a holomorphic germ with an isolated critical point at the origin, and let be the negative gradient flow of with respect to a Riemannian metric. We show that the origin is an isolated invariant set of and that its Conley index is the suspension of the Milnor fiber, so that : the Milnor number is the rank of the homological Conley index of the flow in degree . The proof uses continuation to a holomorphic morsification of . Since is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the nondegenerate rest points that result, and the sum property of the Conley index applies. McCord's theorem then recovers the Poincaré–Hopf index of . [Source notation: unrecognized commands C, CH, Z, im, rank, re; their literal command names are retained; no definitions are inferred.]
A math.GN paper on universal spaces and hyperspaces; descriptive-set-theoretic rather than continuum-theoretic, included in case the hyperspace complexity is of interest.
TL;DR
There 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.
Problem
Construct a universal quasi-Polish space and determine the complexity of the Polish subspaces in its hyperspace.
Main result
There is a quasi-Polish space such that every quasi-Polish space is homeomorphic to a closed subspace of ; the hyperspace is quasi-Polish. The collection of Polish closed subspaces of is coanalytic and not Borel in ; in particular it is -complete.
Methods / framework
Not specified in the abstract.
Context
Descriptive set theory, quasi-Polish spaces and hyperspaces.
In this paper, we first construct a universal quasi-Polish space Y with the property that every quasi-Polish space is homeomorphic to a closed subspace of Y. This gives a hyperspace F(Y) of all quasi-Polish spaces which is itself a quasi-Polish space. We then prove that the collection of all Polish closed subspaces of Y forms a coanalytic, non-Borel subset of F(Y). In particular, this collection is -complete.
complex dynamicshyperspaces and descriptive topology
Relevance
Continuity of Julia sets in the Hausdorff topology is a hyperspace statement about invariant compacta.
TL;DR
Analogues 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.
Problem
Study continuity of Julia sets for families of one-variable rational maps over non-archimedean fields and with the hybrid norm.
Main result
In both the non-archimedean (Berkovich) and the hybrid settings the author proves an analogue of the Mañé–Sad–Sullivan theorem on continuity of the Julia set in the Hausdorff topology. The non-archimedean case uses a maximum modulus principle, proved for analytic functions on analytic spaces without boundary.
Methods / framework
A maximum modulus principle for analytic functions over analytic spaces without boundary.
Context
Non-archimedean and hybrid complex dynamics; Hausdorff continuity of Julia sets.
Let be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over and over equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.
Variational principles for an entropy notion defined via orbit segments; topological entropy theory.
TL;DR
The 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.
Problem
Study -neutralized Pfister–Sullivan entropy, which measures the complexity of orbit segments whose empirical measures lie near a given invariant measure.
Main result
A variational conjecture of Dong and Qiao is disproved. Variational formulas over probability measures are obtained for upper and lower -neutralized topological entropies, with new measure-theoretic quantities via PS localization. Upper bounds under smoothness assumptions and lower bounds for diffeomorphisms are given; for and Anosov diffeomorphisms the -neutralized PS entropy equals for every ergodic . Exact formulas hold on mixing surface hyperbolic basic sets, and the whole-manifold formula can fail for surface diffeomorphisms.
Methods / framework
Not specified in the abstract.
Context
Entropy theory, variational principles and smooth ergodic theory.
In this paper, we study -neutralized Pfister–Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C. Dong and Q. Qiao [On -neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower -neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper -neutralized topological entropy (iii) We obtain upper bounds for -neutralized PS entropy under different smoothness assumptions and lower bounds for diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For diffeomorphisms and Anosov diffeomorphisms, the -neutralized PS entropies equal for every ergodic measure . We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for surface diffeomorphisms.
Expanding Markov maps of the interval, including tent maps and beta-transformations, are one-dimensional dynamics; the results are dimension-theoretic.
TL;DR
For 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.
Problem
Compute Hausdorff dimensions of intersections of quantitative recurrence sets with dynamically badly approximable sets.
Main result
For a class of expanding Markov maps on the unit interval including sofic -transformations, Gauss maps, tent maps and cookie-cutter maps, the Hausdorff dimension of the intersection of a recurrence set with a badly approximable set is determined by the same pressure function as the recurrence set alone. Dimensions are also obtained for recurrence sets intersected with non-recurrent sets and for shrinking target sets intersected with badly approximable sets.
Methods / framework
Pressure functions.
Context
Quantitative recurrence, shrinking targets and dimension theory for interval maps.
The setting is expanding Markov maps (EMMs) of the interval with finite or countable alphabet, satisfying uniform expansion, a weak distortion condition, a mixing condition and a Markov (or, for countable alphabets, full-image) property; examples are sofic -transformations, the Gauss map, tent maps and cookie-cutter maps. All sets live on the repeller . For a positive continuous , the recurrence set consists of with infinitely often, and of whose orbit eventually stays a positive distance from .
Theorems 1.1–1.2: , equal to the root of for finite alphabets and to for countable ones. Theorems 1.3–1.4 give the analogue for radii , with replacing ; for finite alphabets the dimension is when , a case left open for countable alphabets. Theorem 1.5: if , both and are empty; otherwise both have the full dimension .
The upper bounds are immediate from inclusions; the work is in the lower bounds, obtained from Cantor-type subsets built from free blocks (which carry the pressure), returning blocks (forcing returns at sparse prescribed times) and safe blocks (keeping the orbit away from ). The authors stress why this is not automatic: in the -transformation case is meagre, so it is not in any Falconer large-intersection class and its intersection with recurrence sets cannot be read off from countable-intersection properties.
We study a class of expanding Markov maps on the unit interval, which includes sofic -transformations, Gauss maps, tent maps and cookie-cutter maps. We investigate the quantitative recurrence property for points that are dynamically badly approximable, namely, points whose orbits eventually remain a positive distance from a given point. We show that the Hausdorff dimension of the intersection of a recurrence set and a badly approximable set is determined by the same pressure function as that of the corresponding recurrence set. We further obtain the Hausdorff dimensions of two related intersections: recurrence sets with non-recurrent sets, and shrinking target sets with badly approximable sets.
Existence and non-existence of closed orbits for flows transverse to foliations, including C^0-small modifications and plug constructions; topology of flows.
TL;DR
The 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 -small changes.
Problem
Determine which foliations force every transverse flow to have a closed orbit.
Main result
The property is fully characterized among finite depth foliations on atoroidal 3-manifolds. For many foliations, a -small modification of the foliation breaks it.
Methods / framework
Endperiodic maps to produce closed orbits. · Dynamical plugs inserted along curves of attracting holonomy to build flows without closed orbits.
Context
Foliations, flows on 3-manifolds and closed orbits.
Throughout, is a closed, connected, oriented, atoroidal 3-manifold and a 2-dimensional foliation; "forces closed orbits" if every transverse flow has a closed orbit. Theorem A characterizes this for finite-depth foliations: forces closed orbits if and only if it contains a locally top-depth region that is not a topological product, or is -approximated by such foliations (equivalently, it has "connecting junctures"); if not, it admits a transverse flow with no closed orbits. Corollary B: a finite-depth foliation that does not force closed orbits is an iterated shear of one that does. Corollary C: a foliation without holonomy forces closed orbits.
Theorem D (instability): if every leaf of has genus, is -approximated by foliations that do not force closed orbits, so the property is a property of the manifold only when it admits no taut foliation. The authors observe that the closed orbits that are forced always come from periodic points of induced (approximate) monodromies, guaranteed by the Lefschetz fixed point theorem on the universal cover of a leaf.
On the forcing side the tools are endperiodic maps (Handel–Miller theory as written by Cantwell–Conlon–Fenley, and Landry–Minsky–Taylor) and approximation by fibrations over the circle, in which case every transverse flow is a suspension flow. On the other side, a key lemma (Lemma 4.2) inserts a modified Schweitzer plug transverse to the foliation along nonseparating curves with two-sided attracting holonomy; a stated novelty is producing aperiodic flows with few plugs, sometimes one. The hypothesis enters only through the Cantwell–Conlon criterion that all junctures are compactly supported.
We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a -small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.
In dimension two the proof uses invariant annuli from low-dimensional dynamics and C^0 symplectic topology; surface dynamics with a topological component.
TL;DR
Every 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.
Problem
Polterovich's 2002 question on the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two, and Hofer recurrence in higher dimensions.
Main result
On surfaces of genus at least one, every non-trivial Hamiltonian diffeomorphism has all positive iterations separated from the identity in Hofer's metric. In higher dimensions, autonomous Hamiltonian flows on symplectically hyperbolic manifolds are Hofer non-recurrent, and Hofer recurrence is characterized for functions of the moment polytope on monotone toric manifolds. The results also hold for Viterbo's spectral metric, settling cases of the -rigidity conjecture, and in many cases for the -metric.
Methods / framework
Invariant annuli from low-dimensional dynamics. · symplectic topology for approximately invariant Lagrangian submanifolds. · Quantitative Lagrangian Floer theory.
Context
Symplectic topology, Hofer geometry and surface dynamics.
We solve a well-known question of Polterovich from 2002 regarding the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two: every non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterations separated from the identity in Hofer's metric. In higher dimensions, we make progress on the case of autonomous Hamiltonian flows: we prove Hofer non-recurrence for symplectically hyperbolic manifolds, and completely characterize Hofer recurrence for functions of the moment polytope on monotone toric manifolds. These results apply equally well to Viterbo's spectral metric, settling cases of the -rigidity conjecture, and therefore in many cases also to the -metric. Our approach relies on the philosophy that symplectically visible invariant sets govern symplectic non-recurrence phenomena. In dimension two, we use invariant annuli provided by low-dimensional dynamics, and apply methods of symplectic topology to approximately invariant Lagrangian submanifolds. For symplectically hyperbolic manifolds, the invariant sets are provided by sublevel and superlevel sets of autonomous Hamiltonians, while in the toric case, they comprise Lagrangian torus fibers. Across the arguments, we use quantitative Lagrangian Floer theory and its relation to notions of classical dynamics.
flows and foliationssmooth and hyperbolic dynamics
Relevance
Topological and combinatorial study of pseudo-Anosov flows and their closed orbits on 3-manifolds.
TL;DR
An algorithm drills closed orbits out of transitive pseudo-Anosov flows at the level of veering triangulations, giving computational evidence for a conjecture of Ghys.
Problem
Combinatorialize drilling closed orbits of transitive pseudo-Anosov flows via veering triangulations.
Main result
An algorithm takes a veering triangulation and a flow cycle and outputs a veering parent, combinatorializing drilling. Implemented, it produces layered parents of non-layered veering triangulations (hence Birkhoff sections) and geometric parents of non-geometric ones, finds closed orbits not ambiently isotopic to the geodesics in their free homotopy classes, and shows certain veering triangulations encode almost orbit equivalent flows, giving computational evidence for Ghys' conjecture that transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.
Methods / framework
Local approximation of the loom space of a veering triangulation. · A modification of the Agol–Guéritaud construction.
Context
Pseudo-Anosov flows, veering triangulations and 3-manifold topology.
We develop an algorithm that takes as input a veering triangulation V and a flow cycle c of V, and outputs a veering parent V_c of V. Constructing veering parents combinatorializes drilling out closed orbits of transitive pseudo-Anosov flows. The algorithm relies on locally approximating the structure of the loom space associated to V in sufficient detail to carry out a certain modification of the Agol-Guéritaud construction. We use an implementation of the drilling algorithm to construct layered parents of non-layered veering triangulations (and hence Birkhoff sections for transitive pseudo-Anosov flows) and geometric parents of non-geometric veering triangulations. We also find closed orbits of (drilled) pseudo-Anosov flows that are not ambiently isotopic to the geodesics in their free homotopy classes, and show that certain veering triangulations encode almost orbit equivalent flows. In particular, we provide new computational evidence for a conjecture of Ghys asserting that all transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.
, the model space for cubic capture straightening in the Inou–Kiwi framework.
with hyperbolic 
. For non-degenerate pairs with 
, shrinking of both planar puzzles implies local connectedness at 
real parabolic and 
, although both the parameter space and the filled Julia fibre are locally connected there.
, the non-escaping locus of the skew product 
, and its restriction 
over the Mandelbrot set. Theorem 1.1: 
is closed, connected and completely invariant, the projection to the parameter is proper, and 
, and is locally connected at every 
: 
outside the main cardioid and neither the critical orbit nor the orbit of 
hitting the dividing fixed point 
. The author builds mixed puzzle pieces by moving Yoccoz puzzle pieces holomorphically over Roesch's parapuzzle pieces (Słodkowski extension), and proves that the mixed pieces shrink to the point exactly when both the parapuzzle and the dynamical puzzle shrink; in that case 
.
there is 
where 
and 
are both locally connected there. The proof uses Lavaurs limits from parabolic implosion together with a "gate" lemma: real parabolic roots separate the real slice of 
. Theorem 1.5 gives 
from Shishikura's results and a Fubini-type slicing inequality of Héra–Keleti–Máthé. The motivation is the Inou–Kiwi capture straightening map, whose model space is 
, we study the non-escaping locus 
, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here 
is the Mandelbrot set, 
is the filled Julia set of 
, and 
. We prove that 
has full Hausdorff dimension 
. The set 
is locally connected at every 
. At parameters on 
, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of 
, with Conley index the suspension of the Milnor fibre.
, and its Conley index is homotopy equivalent to the suspension of the Milnor fibre, a wedge of 
spheres 
. Hence the Milnor number equals the rank of the homological Conley index in degree 
, and the Poincaré–Hopf index of 
is recovered.
as a first integral, ruling out connecting orbits after a rotation. · The sum property of the Conley index and McCord's theorem.
be a holomorphic germ with an isolated critical point at the origin, and let 
with respect to a Riemannian metric. We show that the origin is an isolated invariant set of 
: the Milnor number is the rank of the homological Conley index of the flow in degree 
is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the 
. [Source notation: unrecognized commands C, CH, Z, im, rank, re; their literal command names are retained; no definitions are inferred.]
such that every quasi-Polish space is homeomorphic to a closed subspace of 
is quasi-Polish. The collection of Polish closed subspaces of 
-complete.
-complete.
be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over 
equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.
-Neutralized Pfister-Sullivan Entropy
diffeomorphisms are given; for 
and 
Anosov diffeomorphisms the 
for every ergodic 
. Exact formulas hold on mixing surface hyperbolic basic sets, and the whole-manifold formula can fail for 
surface diffeomorphisms.
Anosov diffeomorphisms, the 
-transformations, Gauss maps, tent maps and cookie-cutter maps, the Hausdorff dimension of the intersection of a recurrence set with a badly approximable set is determined by the same pressure function as the recurrence set alone. Dimensions are also obtained for recurrence sets intersected with non-recurrent sets and for shrinking target sets intersected with badly approximable sets.
distortion condition, a mixing condition and a Markov (or, for countable alphabets, full-image) property; examples are sofic 
. For a positive continuous 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consists of 
with 
infinitely often, and 
of 
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, equal to the root 
of 
for finite alphabets and to 
for countable ones. Theorems 1.3–1.4 give the analogue for radii 
, with 
replacing 
when 
, a case left open for countable alphabets. Theorem 1.5: if 
, both 
and 
are empty; otherwise both have the full dimension 
.
is meagre, so it is not in any Falconer large-intersection class and its intersection with recurrence sets cannot be read off from countable-intersection properties.
-small changes.
is a closed, connected, oriented, atoroidal 3-manifold and 
a 2-dimensional foliation; 
-rigidity conjecture, and in many cases for the 
-invariant
-actions