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The tent map family is arguably the simplest one-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works, the second author, jointly with J. Yorke, studied the structural graph and backward limits of S-unimodal maps. In this article, we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting periodic orbits and whose structural graph has a finite number of nodes. This article was started under grant DMS-1832126 and then completed and published under grant DMS-2308225.
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Backward Asymptotics in S-Unimodal Maps
While the forward trajectory of a point in a discrete dynamical system is always unique, in general, a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through x was called by Hero the “special α-limit” (sα-limit for short) of x. In this article, we show that there is a hierarchy of sα-limits of points under iterations of a S-unimodal map: the size of the sα-limit of a point increases monotonically as the point gets closer and closer to the attractor. The sα-limit of any point of the attractor is the whole nonwandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.
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- Award ID(s):
- 1832126
- PAR ID:
- 10327920
- Date Published:
- Journal Name:
- International journal of bifurcation and chaos in applied sciences and engineering
- Volume:
- 32
- Issue:
- 6
- ISSN:
- 1793-6551
- Page Range / eLocation ID:
- 2230013
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0218127422300130
