On distributional spectrum of piecewise monotonic maps

Periodic structures for nonlinear piecewise contracting maps

In this paper, we first show that any nonlinear monotonic increasing contracting maps with one discontinuous point on a unit interval which has an unique periodic point with period $n$ conjugates to a piecewise linear contracting map which has periodic point with same period. Second, we consider one parameter family of monotonic increasing contracting maps, and show that the family has the periodic structure called Arnold tongue for the parameter which is associated with the Farey series. This implies that there exist a parameter set with a positive Lebesgue measure such that the map has a per…

On the attractor of piecewise expanding maps of the interval

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary proof of the density of periodic orbits.

Dimension groups for interval maps II: the transitive case

Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.

Monotonicity of entropy and positively oriented transversality for families of interval maps

In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.

区间上分段连续映射的拓扑熵概念的统一

Unifying topological entropy notions for piecewise continuous maps on the interval

本研究推广了AKM拓扑熵定义以适用于紧区间分段连续自映射，证明其可拓展原AKM熵性质、与Bowen公式度量无关且对分段严格单调映射等价于Misiurewicz-Szlenk公式，统一了紧空间下该类映射的常用熵概念。

A Continuous Family of Non-Monotonic Toral Mixing Maps

We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the `stretching and folding' action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well known systems, Arnold's Cat Map and a map due to Cerbelli and Giona, both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such fin…

Parry's topological transitivity and f-expansions

In his 1964 paper on f-expansions, Parry studied piecewise-continuous, piecewise-monotonic maps F of the interval [0,1), and introduced a notion of topological transitivity different from any of the modern definitions. This notion, which we call Parry topological transitivity, (PTT) is that the backward orbit O^-(x)={y:x=F^ny for some n\ge 0} of some x\in[0,1) is dense. We take topological transitivity (TT) to mean that some $x$ has a dense forward orbit. Parry's application to f-expansions is that PTT implies the partition of [0,1) into the "fibers" of F is a generating partition (i.e., f-exp…

Entropy-continuity for interval maps with holes

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole with uniform expansion. We show that under suitable conditions the topological entropy varies locally Hoelder continuously with the local Hoelder exponent depending itself on the value of the topological entropy.

The almost sure invariance principle for unbounded functions of expanding maps

We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: - For uniformly expanding maps, they are square integrable with respect to the…

Dimension groups for interval maps

With each piecewise monotonic map of the unit interval, a dimension triple is associated. The dimension triple, viewed as a Z[t, t^{-1}] module, is finitely generated, and generators are identified. Dimension groups are computed for Markov maps, unimodal maps, multimodal maps, and interval exchange maps. It is shown that the dimension group defined here is isomorphic to K_0(A), where A is a C*-algebra (an "AI-algebra") defined in dynamical terms.