Real polynomial diffeomorphisms with maximal entropy: Tangencies

By Eric Bedford, J. Smillie. 2001

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Like the Henon diffeomorphisms of R2, the quadratic maps of R, have also played a central role in the field of dynamical systems. These two families of dynamical systems fit together naturally, which raises the question of the extent to which the dynamics is similar. One difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms. Substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter. Despite significant recent progress in the theory of Henon diffeomorphisms, due to Benedicks and Carleson and many others, there are still many phenomena that are not nearly so well understood in this two-dimensional setting as they are for quadratic maps. One phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters. In one dimension the nature of this dependence is understood, and the answer is summarized by the principle of monotonicity. Loosely stated, this is the assertion that the complexity of fa does not

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/beb0ff07a59acdbc23caece404c701e64b474f0f

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Citations in the corpus

Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies (2003, background): “D S] 2 N ov”
Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences (2006, background): “In case (ii), f is a real map of maximal entropy in the sense of [2] and [7].”
Dynamics of automorphisms of compact complex surfaces (2014, unknown): “While most results described in the previous paragraphs concern the stochastic properties of the dynamics of automorphisms, there are important results concerning topological aspects of the dynamics of automorphisms of C2 (see [83, 84], [82], and [12, 13, 14] for example).”