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Smooth deformations of piecewise expanding unimodal maps

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  • 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 $k\ge 2$.
    Mathematics Subject Classification: Primary: 37E05, 37C15; Secondary: 37F15, 37A10, 37C20, 32G15, 58H15.


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