Smooth deformations of piecewise expanding unimodal maps

Viviane Baladi; Daniel Smania

doi:10.3934/dcds.2009.23.685

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2009, Volume 23, Issue 3: 685-703. Doi: 10.3934/dcds.2009.23.685

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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
2.
Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970

Received: November 30, 2007

Revised: June 30, 2008

Published: August 2009

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Abstract

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 $k\ge 2$.

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Mathematics Subject Classification: Primary: 37E05, 37C15; Secondary: 37F15, 37A10, 37C20, 32G15, 58H15.

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