Scrambled Edges in Dynamics, Graphs & Topological Matter

• Scrambled Edges are defined as disruptions in the original order of geometric, dynamical, or informational structures, with a distinct meaning in each field.
• In topological dynamics, they refer to systems where every pair of distinct points forms a Li–Yorke pair, achieved via graph-cover constructions and inverse limit techniques.
• Applications range from mitigating clutter in near-planar graph drawings to enabling coherent edge transport in topological quantum materials and novel image scrambling methods.

Searching arXiv for the cited works to ground the article in current bibliographic records. arxiv_search.query({"^^^^^^^^8search_query8 OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^^^^^"," to=arxiv_search.query code blocks Refining the search by querying representative titles and IDs individually. arxiv_search.query({"^^^^^^^^8search_query8 Completely Scrambled System Exists\" OR ^^^^^^^^8ti:\8 construction of a completely scrambled system by graph covers\" OR ^^^^^^^^8ti:\8 scrambled sets, uniform rigidity and weak mixing\"","^^^^^^^^8start8 OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^}) to=arxiv_search.query code blocks arxiv_search.query({"^^^^^^^^8search_query8 to=arxiv_search.query code blocks “Scrambled edges” is not a single standardized term across the arXiv literature. Instead, it recurs in several technically distinct settings in which an “edge” or edge-like object is structurally disruptive, dynamically exceptional, or deliberately rearranged. In topological dynamics, the relevant notion is a completely scrambled system, where every two distinct points form a Li–Yorke pair; a decisive result is the existence of a topologically mixing homeomorphism on the Cantor set that is completely scrambled (&&&^^^^8search_query8^^^^&&&). In graph drawing, the analogous objects are cluttering edges in near-planar graphs, whose geometric influence obscures the planar backbone under spring-based layouts (&&&^^^^8max_results8^^^^&&&). In two-dimensional topological materials, edge modes alter operator growth so that information on the boundary is transported without fully scrambling, producing dynamical scarring in out-of-time-ordered correlators (&&&^^^^8search_query8^^^^&&&). Related uses appear in genomics, signal processing, entanglement detection, image scrambling, and geometric unfolding, where scrambling acts on segment order, frequency order, outcome labels, pixel coordinates, or cut edges rather than on a planar graph edge in the graph-theoretic sense (&&&^^^^8ti:\8^^^^&&&, &&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8start8^^^^&&&, Simnacher et al., 2019, &&&^^^^8 OR ti:\8^^^^&&&, &&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^&&&).

^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^. Conceptual range of the term

Across these literatures, the word “scrambled” consistently denotes loss of an original local ordering, while “edge” denotes either a geometric boundary, a graph edge, an edge state, or a cut edge. The resulting objects are not interchangeable. In topological dynamics, scrambling is defined through proximality and failure of asymptoticity. For a dynamical system PRESERVED_PLACEHOLDER_^^^^8search_query8^^^^, a pair PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ is proximal if

PRESERVED_PLACEHOLDER_^^^^8start8^^^^

and asymptotic if

PRESERVED_PLACEHOLDER_^^^^8max_results8^^^^

A Li–Yorke pair is proximal but not asymptotic, and a system is completely scrambled when every two distinct points form such a pair (&&&^^^^8search_query8^^^^&&&).

In graph drawing, the emphasis is not on orbit complexity but on readability. A cluttering edge is an edge whose presence strongly distorts a spring layout because it connects vertices that are not already locally close in the graph’s structure. The central problem is the visibility of a planar backbone inside a nearly planar graph, not combinatorial nonplanarity per se (&&&^^^^8max_results8^^^^&&&).

In two-dimensional topological matter, scrambling is measured by out-of-time-ordered correlators rather than by topological transitivity or graph layout. There, edge modes do not function as disruptive edges; instead, they support long-lived, boundary-confined transport of perturbation information. This suggests that “scrambled edges” can denote either edges that destroy structure, as in graph drawing, or edges that resist full scrambling, as in topological phases (&&&^^^^8search_query8^^^^&&&).

^^^^8start8^^^^. Completely scrambled dynamics and graph-cover constructions

The most developed mathematical use of scrambling in the supplied literature lies in topological dynamics. “Mixing Completely Scrambled System Exists” proves that there exists a mixing homeomorphism on the Cantor set which is completely scrambled, thereby solving a ^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8ti:\8^^^^ year old open problem left open since Huang–Ye’s ^^^^8start8search_query8search_query8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ paper (&&&^^^^8search_query8^^^^&&&). The construction proceeds by first building a zero-dimensional system via inverse limits of graph covers, then proving that it is mixing and has no asymptotic pairs except the trivial one, and finally passing to higher-dimensional transitive and weakly mixing examples by suspension and factor constructions (&&&^^^^8search_query8^^^^&&&).

The zero-dimensional model is built from a sequence of finite directed graphs and bidirectional covers

PRESERVED_PLACEHOLDER_^^^^8search_query8^^^^

with inverse limit

PRESERVED_PLACEHOLDER_^^^^8ti:\8^^^^

The edge relation induces a homeomorphism PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8^^^^ on the inverse limit. Each graph contains a distinguished vertex PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8^^^^, a distinguished loop $e_{n,0}$, and cycles $c_{n,1},\dots,c_{n,n}$ of increasing length. The bonding maps send the first and last edges of higher-level cycles to the special loop, while mapping the interior to long concatenations of lower-level cycles. The resulting system has a unique fixed point

PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^

and the paper proves that PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ is the only periodic point, every point has PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8start8^^^^ in its PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8max_results8^^^^-limit set, the system is proximal, and the system is topologically mixing (&&&^^^^8search_query8^^^^&&&).

The obstruction to nontrivial asymptotic pairs is encoded by a degree function

PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^

This degree is invariant under the dynamics. The paper shows that if one point has finite degree PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8ti:\8^^^^, then any asymptotic partner must have degree at most PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8 OR ti:\8^^^^; if the degrees differ by exactly one, the pair cannot be asymptotic; and if two points have the same finite degree and are asymptotic, then they must be equal. Combined with proximality, this yields complete scrambling (&&&^^^^8search_query8^^^^&&&).

The graph-cover approach itself had already been developed in “The construction of a completely scrambled system by graph covers,” where the inverse limit of directed graph covers was used to construct a transitive, completely scrambled, ^^^^8search_query8^^^^-dimensional system that is not locally equicontinuous (&&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^&&&). A related structural result, “Invariant scrambled sets, uniform rigidity and weak mixing,” showed that a non-trivial transitive system has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and a dense Mycielski invariant PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8 OR ti:\8^^^^-scrambled set for some PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)88^^^^ if and only if it has a fixed point and is not uniformly rigid. The same paper also provided methods for constructing completely scrambled systems that are weakly mixing, proximal, and uniformly rigid (&&&^^^^8start8^^^^&&&). “Syndetic proximality and scrambled sets” further distinguished ordinary scrambled pairs from syndetically scrambled pairs, showing that syndetic proximality is much more rigid than ordinary proximality and that the two notions coincide only in special classes (&&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^&&&).

A common misconception in this area is that complete scrambling is necessarily incompatible with strong recurrence or mixing. The Cantor-set example shows the opposite: complete scrambling can coexist with topological mixing, and higher-dimensional completely scrambled examples can be made transitive or weakly mixing (&&&^^^^8search_query8^^^^&&&).

^^^^8max_results8^^^^. Near-planar graph drawing and cluttering edges

In graph drawing, the closest direct analogue to “scrambled edges” is the notion of cluttering edges introduced for nearly planar graphs. The problem arises because planar drawings are usually preferred, planarity testing and planar layout are linear-time tasks, but readable drawings of nearly planar graphs remain difficult. Standard force-directed methods can cause extra edges to shorten graph distances that should remain large in the planar backbone, yielding inward folding, outer-face compression, and many crossings (&&&^^^^8max_results8^^^^&&&).

The key distinction is that not every nonplanar edge is equally harmful. The paper’s working intuition is that if two edge endpoints are already connected by several short alternative routes, then the direct edge is unlikely to damage readability. If instead the endpoints are connected only by relatively long routes, forcing the direct edge to be short can collapse the layout. To proxy the intractable set of all alternative paths, the paper defines the footprint of an edge PRESERVED_PLACEHOLDER_^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^9 as

PRESERVED_PLACEHOLDER_^^^^8start8search_query8^^^^

where the PRESERVED_PLACEHOLDER_^^^^8start8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ are the lengths of the vertex-disjoint paths between PRESERVED_PLACEHOLDER_^^^^8start8start8^^^^ and PRESERVED_PLACEHOLDER_^^^^8start8max_results8^^^^ in PRESERVED_PLACEHOLDER_^^^^8start8search_query8^^^^, sorted non-decreasingly. Computing footprints for all edges has complexity

PRESERVED_PLACEHOLDER_^^^^8start8ti:\8^^^^

because vertex-disjoint paths are computed via a max-flow reduction using the Edmonds–Karp algorithm (&&&^^^^8max_results8^^^^&&&).

Because footprints have variable length, they are normalized to a common dimension PRESERVED_PLACEHOLDER_^^^^8start8 OR ti:\8^^^^ by padding or truncation with a summary statistic PRESERVED_PLACEHOLDER_^^^^8start8 OR ti:\8^^^^. The resulting vectors are treated as data points for Isolation Forest, with complexity

PRESERVED_PLACEHOLDER_^^^^8start88^^^^

where PRESERVED_PLACEHOLDER_^^^^8start8^^^^9 is the number of repetitions, PRESERVED_PLACEHOLDER_^^^^8max_results8search_query8^^^^ the number of edges, and PRESERVED_PLACEHOLDER_^^^^8max_results8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ the number of partitions needed to isolate a point. Once cluttering edges are identified, their weights are set by

PRESERVED_PLACEHOLDER_^^^^8max_results8start8^^^^

yielding heuristic variants PRESERVED_PLACEHOLDER_^^^^8max_results8max_results8^^^^, PRESERVED_PLACEHOLDER_^^^^8max_results8search_query8^^^^, and PRESERVED_PLACEHOLDER_^^^^8max_results8ti:\8^^^^. In the motivating example, explicitly reducing the weights of added nonplanar edges to PRESERVED_PLACEHOLDER_^^^^8max_results8 OR ti:\8^^^^ makes the grid structure much clearer (&&&^^^^8max_results8^^^^&&&).

The method is evaluated with ForceAtlas^^^^8start8^^^^ and Stress Majorization on augmented grids, augmented triangulations, deep triangulations, and a subset of Rome graphs. The reported quality criteria are crossing number PRESERVED_PLACEHOLDER_^^^^8max_results8 OR ti:\8^^^^, angular resolution PRESERVED_PLACEHOLDER_^^^^8max_results88^^^^, crossing resolution PRESERVED_PLACEHOLDER_^^^^8max_results8^^^^9, and Procrustes Statistic PRESERVED_PLACEHOLDER_^^^^8search_query8search_query8^^^^. The heuristic improves augmented grids and triangulations, remains promising but challenging for deep triangulations, and does not help much on Rome graphs, which lack the dense planar substructure the method is designed to recover (&&&^^^^8max_results8^^^^&&&).

An important correction to a simplistic crossing-based view is explicit in the experiments: cluttering edges are not only structure-destroying augmenting edges. Some edges near the outer face can also clutter the layout by contributing to inward folding (&&&^^^^8max_results8^^^^&&&).

^^^^8search_query8^^^^. Boundary transport, OTOCs, and dynamical scarring

In two-dimensional topological materials, scrambling is probed by out-of-time-ordered commutators rather than by graph geometry or Li–Yorke chaos. The basic observable is

PRESERVED_PLACEHOLDER_^^^^8search_query8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^

with

PRESERVED_PLACEHOLDER_^^^^8search_query8start8^^^^

and related OTOC

PRESERVED_PLACEHOLDER_^^^^8search_query8max_results8^^^^

If PRESERVED_PLACEHOLDER_^^^^8search_query8search_query8^^^^ remains small, the perturbation has not significantly affected site PRESERVED_PLACEHOLDER_^^^^8search_query8ti:\8^^^^; if it grows and spreads, information is said to scramble (&&&^^^^8search_query8^^^^&&&).

The bulk behavior in two dimensions is light-cone-like but anisotropic. The onset time increases with distance from the perturbation, yet equal-distance points can exhibit different arrival times because the butterfly velocity depends on direction. The paper attributes this directional dependence to the lattice geometry and full band structure, not to topology itself. Numerical examples are given in the square-lattice Kitaev/Chern model along directions such as PRESERVED_PLACEHOLDER_^^^^8search_query8 OR ti:\8^^^^, PRESERVED_PLACEHOLDER_^^^^8search_query8 OR ti:\8^^^^, and diagonal directions (&&&^^^^8search_query8^^^^&&&).

The boundary behavior is qualitatively different. In phases with chiral or helical edge modes, perturbations placed on the boundary generate dynamical scars: long-lived, non-scrambled OTOC structures that remain concentrated on the edge and move along it. For a linearly dispersing chiral edge mode with

PRESERVED_PLACEHOLDER_^^^^8search_query88^^^^

the asymptotic edge OTOC takes the form

PRESERVED_PLACEHOLDER_^^^^8search_query8^^^^9

The perturbation is transported ballistically by the protected edge mode rather than scrambling isotropically into the bulk (&&&^^^^8search_query8^^^^&&&).

The scar direction follows chirality or helicity: in the Chern/Kitaev lattice, PRESERVED_PLACEHOLDER_^^^^8ti:\8search_query8^^^^ gives clockwise motion and PRESERVED_PLACEHOLDER_^^^^8ti:\8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^ counter-clockwise motion. The measured scar velocity in the PRESERVED_PLACEHOLDER_^^^^8ti:\8start8^^^^ phase,

PRESERVED_PLACEHOLDER_^^^^8ti:\8max_results8^^^^

is compared with an edge-mode velocity from the band structure,

PRESERVED_PLACEHOLDER_^^^^8ti:\8search_query8^^^^

with the discrepancy attributed to nonlinearity of the lattice dispersion away from very low energies (&&&^^^^8search_query8^^^^&&&). In the Kane–Mele model, two perturbations on the top edge generate two counterpropagating scars that pass through each other rather than scattering or scrambling, consistent with noninteracting topologically protected helical edge modes (&&&^^^^8search_query8^^^^&&&).

A frequent misunderstanding is that edge states necessarily accelerate or intensify scrambling. The reported boundary phenomenon is the reverse: the edge supports coherent, ballistic transport that preserves a structured memory of the initial perturbation over long time scales (&&&^^^^8search_query8^^^^&&&).

^^^^8ti:\8^^^^. Rearranged segments, frequencies, and outcome labels

Outside dynamical systems and graph drawing, scrambling often acts on combinatorial or informational assignments rather than on geometric edges themselves. In the ciliate Oxytricha trifallax, “Graph Based Analysis for Gene Segment Organization In a Scrambled Genome” studies inter-gene arrangements in which MDSs overlap, are nested, or interleave across genes. For each micronuclear contig, the paper defines a directed, colored graph whose vertices are MAC contigs and whose edge label

PRESERVED_PLACEHOLDER_^^^^8ti:\8ti:\8^^^^

records three interaction types: overlapping, containment/subsegment, and interleaving. The graph is embedded via the feature vector

PRESERVED_PLACEHOLDER_^^^^8ti:\8 OR ti:\8^^^^

with PRESERVED_PLACEHOLDER_^^^^8ti:\8 OR ti:\8^^^^, and persistent homology is used to analyze the resulting point cloud. The data reveal star-like structures in which one gene can interleave with, or even contain, all of the segments from fifteen or more other genes, and also examples in which as many as six genes mutually interleave or overlap (&&&^^^^8ti:\8^^^^&&&).

In Fourier analysis, “Divergence of mock and scrambled Fourier series” uses “scrambled” for a rearrangement of exponential functions on Lebesgue measure generated by the recursive frequency sets

PRESERVED_PLACEHOLDER_^^^^8ti:\88^^^^

The associated Dirichlet kernel is

PRESERVED_PLACEHOLDER_^^^^8ti:\8^^^^9

and in the Lebesgue case the Mahler measure

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8search_query8^^^^

controls growth. If PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^, then for every PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8start8^^^^ there exists PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8max_results8^^^^ such that

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8search_query8^^^^

for all sufficiently large PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8ti:\8^^^^. This exponential PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8 OR ti:\8^^^^-growth is much worse than the classical logarithmic bound and implies divergence for some continuous functions (&&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8start8^^^^&&&).

In entanglement detection, “scrambled data” means that the measurements are known but the assignment of observed probabilities to measurement outcomes has been forgotten. For two qubits with measurements PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8 OR ti:\8^^^^ and PRESERVED_PLACEHOLDER_^^^^8 OR ti:\88^^^^, the paper shows that Tsallis-PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8^^^^9 and Rényi-PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8search_query8^^^^ entropies can detect entanglement from scrambled data, while Shannon entropy cannot. The optimal lower boundary in the entropy plane is realized by the state family

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^

and the set of states whose scrambled data can also come from a separable state is shown to be non-convex (Simnacher et al., 2019).

These cases share a common formal pattern: the original object is not deleted, but its readable assignment is permuted. A plausible implication is that “scrambling” here is best understood as an operation on representation rather than on substance.

^^^^8 OR ti:\8^^^^. Image scrambling, locally scrambled dynamics, and literal edge cuts

Several other papers use “scrambled” in still different but structurally related ways. “Classical Shadow Tomography with Locally Scrambled Quantum Dynamics” generalizes classical shadows to local unitary ensembles satisfying

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8start8^^^^

For such locally scrambled ensembles, the reconstruction map depends only on the average subsystem purity, encoded in the second entanglement feature

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8max_results8^^^^

The inverse channel takes the form

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8search_query8^^^^

and approximately locally scrambled finite-time Hamiltonian dynamics give controllably small bias governed by a frame-potential gap (Hu et al., 2021).

In computer vision, “Block-wise Scrambled Image Recognition Using Adaptation Network” studies perceptual hiding rather than cryptographic secrecy. An image is divided into PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8ti:\8^^^^ blocks, block locations are shuffled, pixels within blocks are shuffled independently, and the blocks are concatenated. The proposed ELE scheme uses different pixel-shuffling keys for every block and includes block shuffling, with key-space relation

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8 OR ti:\8^^^^

Recognition is then recovered by an adaptation network with block-wise sub-networks, a learnable pseudo permutation matrix PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8 OR ti:\8^^^^, a pixel shuffle layer, and total loss

PRESERVED_PLACEHOLDER_^^^^8 OR ti:\88^^^^

The paper emphasizes that the method is not full encryption and that the adaptation network could itself leak information about the scrambling process (&&&^^^^8 OR ti:\8^^^^&&&).

In image security, “Sudoku Associated Two Dimensional Bijections for Image Scrambling” constructs coordinate permutations from a Sudoku matrix. Besides row-column and block-grid representations, it defines six Sudoku-associated representations, yielding PRESERVED_PLACEHOLDER_^^^^8 OR ti:\8^^^^9 basic bijections for one reference Sudoku matrix. The method uses a ^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8start8start8^^^^-bit key, bit-plane processing, image shifting, and overlapping edge blocks so that arbitrary image sizes can be scrambled without bandwidth expansion. Here boundary pixels and visible contours are disrupted because the coordinate bijections destroy adjacency relations (&&&^^^^8 OR ti:\8^^^^&&&).

A literal sense of “edge” reappears in geometry. “Edge-Unfolding Polycubes with Orthogonally Convex Layers” proves that any polycube with orthogonally convex layers can be edge unfolded (&&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^&&&). There, an edge unfolding consists of cutting the surface only along cube edges and flattening it into a single, non-overlapping planar piece. The algorithm is layer-by-layer, organized around selected band cells $e_{n,0}$^^^^8search_query8^^^^ and $e_{n,0}$^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8^^^^, visited band segments, and bridges between layers. The theorem is about recovering an un-scrambled planar net from a three-dimensional surface by choosing cut edges correctly, rather than about scrambling edges in the informational sense (&&&^^^^8id:(Boroński et al., 2016) OR id:(Shimomura, 2015) OR id:(Foryś et al., 2014) OR id:(Wageningen et al., 2023) OR id:(Szpara et al., 17 Dec 2025) OR id:(Hajij et al., 2018) OR id:(Wu et al., 2012) OR id:(Madono et al., 2020) OR id:(Simnacher et al., 2019) OR id:(Hu et al., 2021) OR id:(Damian et al., 2024) OR id:(Moothathu et al., 2011) OR id:(Dutkay et al., 2011)8search_query8^^^^&&&).

Taken together, these uses show that “scrambled edges” is best treated as a family resemblance term rather than a unitary concept. In some fields, the edge is the source of disorder; in others, it is the locus where disorder fails to become complete; and in still others, scrambling is an intentional reindexing of pixels, outcomes, frequencies, or segments that changes observability without changing the underlying object.