Kneading sequences for unimodal expanding maps of the interval

LetP andAC be two primary sequences with min{P, AC}≥RLR ^∞,ρ(P) and ρ(AC) be the eigenvalues ofP andAC, respectively. Letf∈C ⁰ (I, I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved thatf has the kneading sequenceK(f)≥(RC) ^*m *P if λ≥(ρ(P))^1/2m, andK(f)>(RC) ^*m*AC*E for any shift maximal sequenceE if λ>(ρ(AC))^1/2m. The value of (ρ(P))^1/2m or (ρ(AC))^1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one. Project supported by the National Natural Science Foundation of China     

C*-algebras associated with interval maps

For each piecewise monotonic map of , we associate a pair of C*-algebras and and calculate their K-groups. The algebra is an AI-algebra. We characterize when and are simple. In those cases, has a unique trace, and is purely infinite with a unique KMS state. In the case that is Markov, these algebras include the Cuntz-Krieger algebras , and the associated AF-algebras . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and -transformations. For the case of interval exchange maps and of -transformations, the C*-algebra coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

     

A characterization of uniquely ergodic interval exchange maps in terms of the Jacobi-Perron algorithm

An interval exchange mapT satisfies the infinite distinct orbit condition if theT-orbits of theT-discontinuities are infinite and distinct. We characterize among the interval exchange maps that satisfy this condition those that are uniquely ergodic by the convergence of an associated multidimensional continued fraction in the sense of Jacobi and Perron.     

Topological attractors of contracting Lorenz maps

We study the non-wandering set of contracting Lorenz maps. We show that if such a map f doesn't have any attracting periodic orbit, then there is a unique topological attractor. Furthermore, we classify the possible kinds of attractors that may occur.     

Bernoulli maps of the interval

The ergodic properties of expanding piecewiseC ² maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness). Partially supported by NSF Grant MCS74-19388 and the Sloan Foundation.     

Bernoulliflows over maps of the interval

We consider special flowsT′ built over shift automorphismϕ with a Holder function. We introduce two properties forϕ (R and WM) s.t. weak-mixing forT′ impliesK ifϕ is R and it implies Bernoulliness ifϕ is WM. We apply this to maps of the interval to show that for the Lorentz Attractor Flow with a natural measure weak-mixing implies Bernoulliness. Research partially supported by NSF Grant MCS74-19388.A01 and the Sloan Foundation.     

Renormalization and conjugacy of piecewise linear Lorenz maps

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric.     

Dynamics of acyclic interval maps

We investigate conditions under which a map f in a possibly non-compact interval is acyclic— the only periodic orbits are fixed points. Several earlier results are generalized to maps with multiple fixed points. The chief tools are convergence results due to Coppel and Sharkovski, and the Schwarzian derivative. Illustrative examples are given and open problems are suggested.

He who can digest a second or third fluxion . . . need not, methinks, be squeamish about any point in divinity. ―Bishop George Berkeley, “The Analyst,” 1734

     

A characterization of hyperbolic rational maps

We give a topological characterization of rational maps with disconnected Julia sets. Our results extend Thurston’s characterization of postcritically finite rational maps. In place of iteration on Teichmüller space, we use quasiconformal surgery and Thurston’s original result.     

A spectral characterization of skeletal maps

We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.     

A characterization of adding machine maps

Let f :X→X be a continuous map of a compact metric space to itself. We prove that f is topologically conjugate to an adding machine map if and only if X is an infinite minimal set for f and each point of X is regularly recurrent. Moreover, if X is an infinite minimal set for f and one point of X is regularly recurrent, then f is semiconjugate to an adding machine map.     

Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

A characterization of the interval distance monotone graphs

A simple connected graph G is said to be interval distance monotone if the interval I(u,v) between any pair of vertices u and v in G induces a distance monotone graph. A?¨der and Aouchiche [Distance monotonicity and a new characterization of hypercubes, Discrete Math. 245 (2002) 55-62] proposed the following conjecture: a graph G is interval distance monotone if and only if each of its intervals is either isomorphic to a path or to a cycle or to a hypercube. In this paper we verify the conjecture.     

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy. In this way, we reduce the study ofC ^r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ^∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.     

Topological sequence entropy for maps of the interval

A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

     

Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials

Hubbard trees are invariant trees connecting the points of thecritical orbits of post-critically finite polynomials. Douadyand Hubbard showed in the Orsay Notes that they encode all combinatorialproperties of the Julia sets. For quadratic polynomials, onecan describe the dynamics as a subshift on two symbols, anditinerary of the critical value is called the kneading sequence.Whereas every (pre)periodic sequence is realized by an abstractHubbard tree (see the authors’ preprint from 2007), notevery such tree is realized by a quadratic polynomial. In thispaper, we give an Admissibility Condition that describes preciselywhich sequences correspond to quadratic polynomials. We identifythe occurrence of the so-called ‘evil branch points’as the sole obstruction to being realizable. We also show howto derive the properties of periodic (branch) points in thetree (their periods, relative positions, number of arms andwhether they are evil or not) from the kneading sequence.