Invariant manifolds for non-differentiable operators
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- by Marco Martens and Liviana Palmisano PDF
- Trans. Amer. Math. Soc. 375 (2022), 1101-1169 Request permission
Abstract:
A general invariant manifold theorem is needed to study the topological classes of smooth dynamical systems. These classes are often invariant under renormalization. The classical invariant manifold theorem cannot be applied, because the renormalization operator for smooth systems is not differentiable and sometimes does not have an attractor. Examples are the renormalization operator for general smooth dynamics, such as unimodal dynamics, circle dynamics, Cherry dynamics, Lorenz dynamics, Hénon dynamics, etc. A general method to construct invariant manifolds of non-differentiable non-linear operators is presented. An application is that the $\mathcal C^{4+\epsilon }$ Fibonacci Cherry maps form a $\mathcal C^1$ codimension one manifold.References
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Additional Information
- Marco Martens
- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York
- MR Author ID: 120380
- Email: marco@math.sunysb.edu
- Liviana Palmisano
- Affiliation: Department of Mathematical Sciences, Durham University, Durham, England
- MR Author ID: 1023538
- Email: liviana.palmisano@durham.ac.uk
- Received by editor(s): November 13, 2019
- Received by editor(s) in revised form: April 7, 2021, and June 9, 2021
- Published electronically: November 29, 2021
- Additional Notes: The first author was partially supported by the NSF grant 1600554 and the second author was partially supported by the Leverhulme Trust through the Leverhulme Prize of C. Ulcigrai and by the Trygger foundation, Project CTS 17:50.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1101-1169
- MSC (2020): Primary 37E10, 37C15, 37E20
- DOI: https://doi.org/10.1090/tran/8493
- MathSciNet review: 4369244