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4 Fundamenta Mathematicae

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

100%

Brucks K. , Buczolich Z.

Fundamenta Mathematicae

2000

tom 165

nr 2

95-123

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.

Adding machines, endpoints, and inverse limit spaces

100%

Alvin L. , Brucks K.

Fundamenta Mathematicae

2010

tom 209

nr 1

81-93

Let f be a unimodal map in the logistic or symmetric tent family whose restriction to the omega limit set of the turning point is topologically conjugate to an adding machine. A combinatoric characterization is provided for endpoints of the inverse limit space (I,f), where I denotes the core of the map.

Subcontinua of inverse limit spaces of unimodal maps

100%

Brucks K. , Bruin H.

Fundamenta Mathematicae

1999

tom 160

nr 3

219-246

We discuss the inverse limit spaces of unimodal interval maps as topological spaces. Based on the combinatorial properties of the unimodal maps, properties of the subcontinua of the inverse limit spaces are studied. Among other results, we give combinatorial conditions for an inverse limit space to have only arc+ray subcontinua as proper (non-trivial) subcontinua. Also, maps are constructed whose inverse limit spaces have the inverse limit spaces of a prescribed set of periodic unimodal maps as subcontinua.

On the ∗-product in kneading theory

70%

Brucks K. , Galeeva R. , Mumbrú P. , Rockmore D. , Tresser C.

tom 152

nr 3

189-209

We discuss a generalization of the *-product in kneading theory to maps with an arbitrary finite number of turning points. This is based on an investigation of the factorization of permutations into products of permutations with some special properties relevant for dynamics on the unit interval.