Geometric pressure for multimodal maps of the interval
Geometric pressure for multimodal maps of the interval
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DOI：
10.1090/memo/1246
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发表时间：
2014-05
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影响因子：
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通讯作者：
F. Przytycki;Juan Rivera-Letelier
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文献类型：
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作者：
F. Przytycki;Juan Rivera-Letelier
本文是作者、S.Smirnov等人在Riemann球面上有理映射迭代的背景下建立的三个理论的区间动力学的对应物：非一致双曲性、几何压力和Nice诱导格式的几个概念的等价性，以及导致热力学形式论结果的方法。我们在广义多峰映射的集合中工作，即从R中的有限个紧区间并到R中具有非平坦临界点的光滑映射f，使得映射f在其极大正向不变集K上是拓扑传递的且具有正拓扑熵。我们证明了f|K的非一致双曲性的几个概念是等价的(包括周期轨道上的一致双曲性、K双曲排斥下的TCE和所有周期轨道、Lyapunov双曲性以及拉回的指数收缩)。我们证明了几种几何压力P(T)的定义，即映射f|_K和势-tlog|f‘|的压力，给出了相同的值(包括周期轨道上的压力、树形压力、变分压力和保形压力)。最后，我们证明了，如果K中的所有周期轨道都是双曲排斥的，则函数P(T)是“凝聚”和“冻结”参数之间的t的实解析，并且对于每个这样的t，存在唯一的平衡(和保形)度量，满足强统计性质。
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps f of a finite union of compact intervals in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of f|_K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). We prove that several definitions of geometric pressure P(t), that is pressure for the map f|_K and the potential - t log |f'|, give the same value (including pressure on periodic orbits, "tree" pressure, variational pressures and conformal pressure). Finally we prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the "condensation" and "freezing" parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.