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S-unimodal Misiurewicz maps with flat critical points

Roland Zweimüller (2004)

Fundamenta Mathematicae

We consider S-unimodal Misiurewicz maps T with a flat critical point c and show that they exhibit ergodic properties analogous to those of interval maps with indifferent fixed (or periodic) points. Specifically, there is a conservative ergodic absolutely continuous σ-finite invariant measure μ, exact up to finite rotations, and in the infinite measure case the system is pointwise dual ergodic with many uniform and Darling-Kac sets. Determining the order of return distributions to suitable reference...

Symbolic dynamics and Lyapunov exponents for Lozi maps

Diogo Baptista, Ricardo Severino (2012)

ESAIM: Proceedings

Building on the kneading theory for Lozi maps introduced by Yutaka Ishii, in 1997, we introduce a symbolic method to compute its largest Lyapunov exponent. We use this method to study the behavior of the largest Lyapunov exponent for the set of points whose forward and backward orbits remain bounded, and find the maximum value that the largest Lyapunov exponent can assume.

The Bernoulli shift as a basic chaotic dynamical system

Kučera, Václav (2019)

Programs and Algorithms of Numerical Mathematics

We give a brief introduction to the Bernoulli shift map as a basic chaotic dynamical system. We give several examples where the iterates of a~mapping can be understood using the Bernoulli shift. Namely, the iteration of real interval maps and iteration of quadratic functions in the complex plain.

The recurrence dimension for piecewise monotonic maps of the interval

Franz Hofbauer (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [ 0 , 1 ] , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...

The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos

Lidong Wang, Gongfu Liao, Zhizhi Chen, Xiaodong Duan (2003)

Annales Polonici Mathematici

We discuss the existence of an uncountable strongly chaotic set of a continuous self-map on a compact metric space. It is proved that if a continuous self-map on a compact metric space has a regular shift invariant set then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.

The topological entropy versus level sets for interval maps

Jozef Bobok (2002)

Studia Mathematica

We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?

The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) c a r d f - 1 ( y ) ≥ 2 for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that c a r d f - 1 ( y ) ≥ m for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Trajectory of the turning point is dense for a co-σ-porous set of tent maps

Karen Brucks, Zoltán Buczolich (2000)

Fundamenta Mathematicae

It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map T a with slope a is dense in the interval of transitivity of T a . We prove that the complement of this set of parameters of full measure is σ-porous.

Transitive sensitive subsystems for interval maps

Sylvie Ruette (2005)

Studia Mathematica

We prove that for continuous interval maps the existence of a non-empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.

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