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This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space C1+Lip symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of C1+Lip symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

World University root node

Department of Mathematics

Fulltext

Journal of Statistical Physics

Renormalization of Symmetric Bimodal Maps with Low Smoothness

The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space of C 2+α unimodal maps, for α &gt; 0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space of C 2 unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called C2+, the failure of hyperbolicity is tamer than in C2. Things get much worse with just a bit less smoothness than C2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy. © 2008 Cambridge University Press.

Ergodic Theory and Dynamical Systems

Chaotic period doubling

International Journal of Dynamics & Control

Stochastically stable chaos for q-deformed unimodal maps

Journal of Difference Equations and Applications

Dynamics of q deformed Ricker map

Pramana Journal of Physics

An improved q-deformed Logistic map and its implications

Chaos, Solitons & Fractals

Dynamics of deformed Henon-like maps

Discrete and Continuous Dynamical Systems

Period tripling and quintupling renormalizations below C^2 space

AIP Conference Proceedings

Repelling Hyperbolic Cantor Set of a Bimodal Map

Topological Entropy of One-dimensional Deformed Maps