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].
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.
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?
It is well known that any continuous piecewise monotone interval map f with positive topological entropy is semiconjugate to some piecewise affine map with constant slope . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.
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 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...
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.
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.
In our note, we prove the result that the Hilbert’s cube equipped with metrics, , cannot be isometrically embedded into .
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.
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.
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.
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