We determine the asymptotic behaviour of the iterates of the Perron-Frobenius operator for specific interval maps with an indifferent fixed point which gives rise to an infinite invariant measure.
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.
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 , 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...
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.
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?
Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that 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.
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 with slope a is dense in the interval of transitivity of . We prove that the complement of this set of parameters of full measure is σ-porous.
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.
Let I be a compact real interval and let f:I → I be continuous. We describe an interval analogy of the irrational circle rotation that occurs as a subsystem of the dynamical system (I,f)-we call it an irrational twist system. Using a coding we show that any irrational twist system is strictly ergodic. We also prove that irrational twist systems exist as subsystems of a large class of systems (I,f) having a cycle of odd period greater than one.