Geodesic
Rigidity theorem for self-affine arcs
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 497 (2021), pp. 18-22

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It has been known for more than a decade that, if a self-similar arc $\gamma$ can be shifted along itself by similarity maps that are arbitrarily close to identity, then $\gamma$ is a straight line segment. We extend this statement to the class of self-affine arcs and prove that each self-affine arc admitting affine shifts that may be arbitrarily close to identity is a segment of a parabola or a straight line.
Keywords: self-affine arc, attractor, weak separation property, rigidity theorem. 
@article{DANMA_2021_497_a3,
     author = {A. V. Tetenov and O. A. Chelkanova},
     title = {Rigidity theorem for self-affine arcs},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {18--22},
     publisher = {mathdoc},
     volume = {497},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_497_a3/}
}
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A. V. Tetenov; O. A. Chelkanova. Rigidity theorem for self-affine arcs. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 497 (2021), pp. 18-22. http://geodesic.mathdoc.fr/item/DANMA_2021_497_a3/