![]() Zeros of the kneading invariant and topological entropy for Lorenz maps. Topological conjugation of Lorenz maps by β-transformations. Thesis, University of Warwick, Coventry, UK, 1979. On the Classification of Measure Preserving Transformations of Lebesgue Spaces. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Mathematics Springer: Berlin/Heidelberg, Germany, 1979 Volume 729, pp. The Lorenz attractor and a related population model. Markov subshifts and realization of β-expansions. The classification of topologically expansive Lorenz maps. Maximal measures for piecewise monotonically increasing transformations on. The entropy of a special overlapping dynamical system. Unique developments in non-integer bases. Almost every number has a continuum of β-expansions. Characterization of the unique expansions 1 = ∑ i = 1 ∞ q − n i and related problems. Local dimensions for the random β-transformation. On the Hausdorff dimension of Bernoulli convolutions. In Topics in Dynamics and Ergodic Theory London Mathematical Society Lecture Note Series Cambridge University Press: Cambridge, UK, 2003 Volume 310, pp. Thesis, Universität Bremen, Bremen, Germany, 2019. Finite and Infinite Rotation Sequences and Beyond. Denseness of Intermediate β-shifts of Finite Type. In Proceedings of the 2002 IEEE International Symposium on Circuits and Systems, Scottsdale, AZ, USA, 26– Volume 2, pp. Beta expansions: A new approach to digitally corrected A/D conversion. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors Applied Mathematical Sciences Springer: New York, NY, USA Berlin, Germany, 1982 Volume 41. Representations for real numbers and their ergodic properties. The authors declare no conflict of interest. ![]() ![]() It is worth noting that the former of these two sets has positive Lebesgue measure and is far from being dense in Δ, see Theorem 6 due to Palmer and Glendinning. In contrast, the structure of the set of ( β, α ) in Δ for which Ω β, α is topologically transitive, with respect to the (left) shift map, is notably different to the set of ( β, α ) in Δ for which Ω β, α is a subshift of finite type. In a second article by Li et al., it was shown that this set of parameters is in fact dense in Δ. These results immediately give us that the set of parameters in Δ which give rise to β-shifts of finite type is countable. The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β-shifts, namely transformations of the form T β, α : x ↦ β x + α mod 1 acting on, where ( β, α ) ∈ Δ is fixed and where Δ ≔, and Li et al. ![]()
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