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July  2009, 23(3): 685-703. doi: 10.3934/dcds.2009.23.685

Smooth deformations of piecewise expanding unimodal maps

Viviane Baladi 1, and  Daniel Smania 2,

1. 

D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris, France
2. 

Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970

Received  December 2007 Revised  July 2008 Published  November 2008

In the space of $C^k$ piecewise expanding unimodal maps, $k\geq 1$, we characterize the $C^1$ smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of $C^{k-1+Lip}$ deformations tangent to every given $C^k$ horizontal direction, for $k\ge 2$.
Keywords: unimodal map, deformation theory, piecewise expanding map, topological class.
Mathematics Subject Classification: Primary: 37E05, 37C15; Secondary: 37F15, 37A10, 37C20, 32G15, 58H15.
Citation: Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
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