Homeomorphic restrictions of unimodal maps

Countable inverse limits of postcritical ω-limit sets of unimodal maps

Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim{[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim {ω(c),f| ω(c) }. In this paper we consider the relationship between the limit complexity of ω(c) and the limit complexity of I. We show that if ω(c) is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. W…

Monotonicity of entropy for unimodal real quadratic rational maps

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.

单峰映射的递归、符号动力学与野吸引子

Recurrence, symbolic dynamics, and wild attractors for unimodal maps

本文通过符号动力学研究临界轨道递归性质与野Cantor吸引子的关系，构造了具有野吸引子的非重整化单峰映射族，并证明了长分支组合与持续递归的不兼容性。

Flexibility of entropies for piecewise expanding unimodal maps

We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is…

塔与布拉特利-维什克系统在斐波那契-like单峰映射中的应用

Towers and Bratteli-Vershik systems in Fibonacci-like unimodal maps

本文研究了斐波那契-like单峰映射中通过塔结构构造康托尔集，并推导了其唯一不变测度的显式公式。

Generalized beta-transformations and the entropy of unimodal maps

Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto βx ~\pmod 1$, where $β>1$, and replacing some of the branches with branches of constant negative slope. If the orbit of 1 is finite, then the map is Markov, and we call beta (which must be an algebraic number) a generalized Parry number. We show that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. We give a characterization of the closure o…