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1
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Forcing relation on minimal interval patterns

100%
Bobok J.
Fundamenta Mathematicae
|
2001
|
tom 169
|
nr 2
161-173
EN
Let ℳ be the set of pairs (T,g) such that T ⊂ ℝ is compact, g: T → T is continuous, g is minimal on T and has a piecewise monotone extension to convT. Two pairs (T,g),(S,f) from ℳ are equivalent if the map h: orb(minT,g) → orb(minS,f) defined for each m ∈ ℕ₀ by $h(g^m(minT)) = f^m(minS)$ is increasing on orb(minT,g). An equivalence class of this relation-a minimal (oriented) pattern A-is exhibited by a continuous interval map f:I → I if there is a set T ⊂ I such that (T,f|T) = (T,f) ∈ A. We define the forcing relation on minimal patterns: A forces B if all continuous interval maps exhibiting A also exhibit B. In Theorem 3.1 we show that for each minimal pattern A there are maps exhibiting only patterns forced by A. Using this result we prove that the forcing relation on minimal patterns is a partial ordering. Our Theorem 3.2 extends the result of [B], where pairs (T,g) with T finite are considered.
2
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The topological entropy versus level sets for interval maps (part II)

100%
Bobok J.
Studia Mathematica
|
2005
|
tom 166
|
nr 1
11-27
EN
Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $card f^{-1}(y) ≥ 2$ for every y ∈ [a,b]; (ii) for some m ∈ {∞,2,3,...} there is a countable set L ⊂ [a,b] such that $card f^{-1}(y) ≥ m$ 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.
3
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Forcing relation on interval patterns

100%
Bobok J.
Fundamenta Mathematicae
|
2005
|
tom 187
|
nr 1
37-60
EN
We consider-without restriction to the piecewise monotone case-a forcing relation on interval (transitive, roof, bottom) patterns. We prove some basic properties of this type of forcing and explain when it is a partial ordering. Finally, we show how our approach relates to the results known from the literature.
4
Content available remote

On entropy of patterns given by interval maps

100%
Bobok J.
Fundamenta Mathematicae
|
1999
|
tom 162
|
nr 1
1-36
EN
Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in[11].
5
Content available remote

Semiconjugacy to a map of a constant slope

100%
Bobok J.
Studia Mathematica
|
2012
|
tom 208
|
nr 3
213-228
EN
It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{top}(f)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{top}(f)}$. We prove this result for a class of Markov countably piecewise monotone continuous interval maps.
6
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The topological entropy versus level sets for interval maps

100%
Bobok J.
Studia Mathematica
|
2002
|
tom 152
|
nr 3
249-261
EN
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?
7
Content available remote

Twist systems on the interval

100%
Bobok J.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 2
97-117
EN
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.
8
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X-minimal patterns and a generalization of Sharkovskiĭ's theorem

64%
Bobok J., Kuchta M.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 1
33-66
EN
We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.
9
Content available remote

Does a billiard orbit determine its (polygonal) table?

64%
Bobok J., Troubetzkoy S.
Fundamenta Mathematicae
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2011
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tom 212
|
nr 2
129-144
EN
We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation under additional regularity conditions on the orbit.
10
Content available remote

Topological entropy on zero-dimensional spaces

64%
Bobok J., Zindulka O.
Fundamenta Mathematicae
|
1999
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tom 162
|
nr 3
233-249
EN
Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈[0,∞] there is a homeomorphism on X whose topological entropy is a.
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