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Equi-topological entropy curves for skew tent maps in the square

  • Zoltán Buczolich EMAIL logo und Gabriella Keszthelyi
Veröffentlicht/Copyright: 30. November 2017
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Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 67 Heft 6

Abstract

We consider skew tent maps Tα, β(x) such that (α,β)∈[0,1]2 is the turning point of TTα, β, that is, Tα, β = βαx for 0≤ x ≤ α and Tα, β(x) = β1−α (1−x) for α < x ≤ 1. We denote by M = K(α,β) the kneading sequence of TTα, β and by h(α,β) its topological entropy. For a given kneading squence M we consider equi-kneading, (or equi-topological entropy, or isentrope) curves (α,φM(α)) such that K(α,φM(α)) = M. To study the behavior of these curves an auxiliary function ΘM(α,β) is introduced. For this function ΘM(α,φM(α)) = 0, but it may happen that for some kneading sequences ΘM(α,β) = 0 for some β < φM(α) with (α,β) still in the dynamically interesting quarter of the unit square. Using ΘM we show that the curves (α,φM(α)) hit the diagonal {(β,β): 0.5 < β < 1} almost perpendicularly if (β,β) is close to (1,1). Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for M = RLLRC the curve (α,φM(α)) is not orthogonal to the diagonal. On the other hand, for M = RLC it is.

With different parametrization properties of equi-kneading maps for skew tent maps were considered by J. C. Marcuard, M. Misiurewicz and E. Visinescu.

MSC 2010: Primary 37B40; Secondary 28D20; 37E05; 37E20
Keywords: skew tent map; topological entropy; level set

Both authors were supported by the Hungarian National Research, Development and Innovation Office – NKFIH, Grant 104178

Dedicated to Professor Paolo de Lucia on the occasion of his 80th birthday

Communicated by Anatolij Dvurečenskij

References

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Received: 2016-4-20
Accepted: 2016-10-18
Published Online: 2017-11-30
Published in Print: 2017-11-27

© 2017 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
https://doi.org/10.1515/ms-2017-0072
Schlagwörter für diesen Artikel
skew tent map; topological entropy; level set

Artikel in diesem Heft

  2. Paolo de Lucia
  4. Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
  6. A note on field-valued measures
  8. Topologies and uniformities on d0-algebras
  10. On some properties of 𝓙-approximately continuous functions
  12. Porous subsets in the space of functions having the Baire property
  14. Rademacher’s theorem in Banach spaces without RNP
  16. Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
  18. Measure games on pseudo-D-lattices
  20. On some properties of k-subadditive lattice group-valued capacities
  22. Lp Spaces in vector lattices and applications
  24. The Choquet integral with respect to fuzzy measures and applications
  26. A note on the range of vector measures
  28. Ideal convergent subsequences and rearrangements for divergent sequences of functions
  30. Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
  32. A generalization of the exponential sampling series and its approximation properties
  34. Monotonicity and total boundedness in spaces of “measurable” functions
  36. Vector lattices in synaptic algebras
  38. Density, ψ-density and continuity
  40. Feather topologies
  42. On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
  44. Rate of convergence of empirical measures for exchangeable sequences
  46. On non-additive probability measures
  48. Equi-topological entropy curves for skew tent maps in the square
  50. On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
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