Regular or stochastic dynamics in real analytic families of unimodal maps

By Artur Avila, M. Lyubich, W. de Melo. 2003

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Generic family with robustly infinitely many sinks (2014, background): “ying the properties above. The main conjecture has been fully proved for one-dimensional maps displaying only one critical point: by Lyubich [Lyu02] for quadratic families, Avila, de Melo and Lyubich [ALdM03] in the analytic case under the hypothesis of negative Schwarz derivative and, Avila and Mor”
Linear response, or else (2014, background): “This condition was first studied for smooth unimodal maps [3, 60].”
The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes (2010, background): “Date: November 3, 2018. 1 2 ARTUR AVILA AND MIKHAIL LYUBICH the proofs of the Regular or Stochastic Dichotomy in the real quadratic family [L4] and more general spaces of quasiquadratic unimodal maps [ALM]. Here we will consider the renormalization operator Rin the space CR d of real unicritical pol”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “…our discussion to dif-feomorphisms, but the same kind of questions, and the Palis conjecture, also apply to endomorphisms—see for example [9, 10, 17, 24, 32, 31] for the setting of one-dimensional maps, [45] in two dimensions, and [5, 6, 40, 41] for the construction of SRB measures in…”