Real bounds, ergodicity and negative Schwarzian for multimodal maps

By S. Strien, E. Vargas. 2004

101 citations recorded by the source corpus.

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained so far only for unimodal maps. The aim of this paper is to provide all the tools for studying general multimodal maps of an interval or a circle, by obtaining * real bounds controlling the geometry of domains of certain first return maps, and providing a new (and we believe much simpler) proof of absense of wandering intervals; * provided certain combinatorial conditions are satisfied, large real bounds implying that certain first return maps are almost linear; * Koebe distortion controlling the distortion of high iterates of the map, and negative Schwarzian derivative for certain return maps (showing that the usual assumption of negative Schwarzian derivative is unnecessary); * control of distortion of certain first return maps; * ergodic properties such as sharp bounds for the number of ergodic components.

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

View Paper: https://www.semanticscholar.org/paper/60309c61ada26af48b3319bd1ecfa664121d030b

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Citations in the corpus

Takens' last problem and existence of non-trivial wandering domains (2015, background): “Also, for unimodal as well as multimodal maps on an interval or a circle, the main difficulty in the their classification in real analytic category was to show the absence of wandering domains, which were developed by many dynamicists [7, 22, 2, 8, 34], see the survey of van Strien [33].”
Periodic attractors of perturbed one-dimensional maps (2013, unknown): “ighbourhood J of csuch that if g∈ F and gn(x) ∈ J for some x∈ N and n≥ 0, then Sgn+1(x) <0. The proof of this theorem follows the same lines as the proof of its single map version, see [Koz00] and [VV04]. One should notice that if all critical points of fare quadratic, one can choose a neighbourh”
Return time statistics via inducing (2000, background): “Recent work of Kozlovski [23], extended to the multimodal case by Van Strien & Vargas [42] shows that the C2 assumption suffices.”
Equilibrium measures for maps with inducing schemes (2006, background): “The negative Schwarzian derivative assumption is not necessary to prove distortion bounds for C 3-unimodal maps with no neutral periodic cycles [21] (and even C 2+η unimodal maps, see [40]), or for C 3 multimodal maps [42].”