Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries
Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries
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DOI：
10.2140/gt.2021.25.111
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发表时间：
2021-01-01
影响因子：
2
通讯作者：
Hall, Toby
中科院分区：
文献类型：
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作者：
Boyland, Philip;de Carvalho, Andre;Hall, Toby
设{f（t）：I -> I}是拓扑熵h（f（t））> 1/2log 2的单峰映射族，（f）over cap（t）：（I）over cap（t）->（I）over cap（t）是它们的自然扩张，其中（I）over cap（t）=（sic）（I，f（t））.在帐篷映射和二次映射所满足的正则性条件下，我们给出了（I）在cap（t）上的Barge-Martin嵌入到球面中的素端的完整刻画.我们还构造了一个球面同胚族{chi（t）：S-2 -> S-2}，它的性质是每个chi（t）是cap（t）上的（f）的因子，通过一个双偶性，使得除了一个纤维外，所有纤维最多包含三个点，并且例外纤维不携带拓扑熵;也就是说，单峰自然扩张实际上是球面同胚。在{f（t）}是帐篷族的情况下，证明了chi（t）是对稠密参数集的广义伪Anosov映射，且f（t）是后临界有限的，使得{chi（t）}是de Carvalho和Hall（Geom. Topol. 8（2004）1127-1188）。
Let {f(t): I -> I} be a family of unimodal maps with topological entropies h(f(t)) > 1/2 log 2, and (f) over cap (t) : (I) over cap (t) -> (I) over cap (t) be their natural extensions, where (I) over cap (t) = (sic) (I, f(t)). Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embeddings of (I) over cap (t) into the sphere. We also construct a family {chi(t) : S-2 -> S-2} of sphere homeomorphisms with the property that each chi(t) is a factor of (f) over cap (t), by a semiconjugacy for which all fibers except one contain at most three points, and for which the exceptional fiber carries no topological entropy; that is, unimodal natural extensions are virtually sphere homeomorphisms. In the case where {f(t)} is the tent family, we show that chi(t) is a generalized pseudo-Anosov map for the dense set of parameters for which f(t) is postcritically finite, so that {chi(t)} is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127-1188).