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Denjoy's anachronistic topological viewpoint on Aubry transition
O Cépas1, G Masbaum2, P Quémerais1
1Institut Néel, CNRS, Université Grenoble Alpes, Grenoble France.
Chaos (Woodbury, N.Y.)
|October 29, 2025
Summary
The Aubry transition, a shift between incommensurate states, is re-explained using Denjoy
Area of Science:
- Physics
- Mathematics
Background:
- The Aubry transition describes a phase transition between incommensurate states.
- It was originally characterized by a "breaking of analyticity."
- Denjoy's work on circle homeomorphisms offers a new perspective.
Purpose of the Study:
- To reframe the Aubry transition using Denjoy's mathematical concepts.
- To connect the breaking of analyticity to topological concepts.
- To provide a new theoretical framework for understanding incommensurate states.
Main Methods:
- Analysis of Denjoy's theory on circle homeomorphisms with irrational rotation numbers.
- Establishing a change of variables from incommensurate ground state variables to phase variables.
- Numerical calculations on the Frenkel-Kontorova model to illustrate the concepts.
Main Results:
- Identified a transformation to simple phase variables exhibiting cyclic order.
- Rephrased the "breaking of analyticity" as a "breaking of topological conjugacy."
- Demonstrated two types of cyclic order based on the nature of the variable change (topological conjugacy vs. semiconjugacy).
Conclusions:
- Denjoy's viewpoint provides a novel mathematical interpretation of the Aubry transition.
- The transition can be understood as a change in topological conjugacy.
- This framework offers deeper insights into the properties of incommensurate systems.
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