Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II

Franz Hofbauer

Studia Mathematica (1993)

  • Volume: 106, Issue: 3, page 213-231
  • ISSN: 0039-3223

Abstract

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We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.

How to cite

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Hofbauer, Franz. "Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II." Studia Mathematica 106.3 (1993): 213-231. <http://eudml.org/doc/216015>.

@article{Hofbauer1993,
abstract = {We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.},
author = {Hofbauer, Franz},
journal = {Studia Mathematica},
keywords = {expanding piecewise monotonic maps; Hausdorff dimension; Hausdorff measure; conformal measures},
language = {eng},
number = {3},
pages = {213-231},
title = {Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II},
url = {http://eudml.org/doc/216015},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Hofbauer, Franz
TI - Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 3
SP - 213
EP - 231
AB - We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.
LA - eng
KW - expanding piecewise monotonic maps; Hausdorff dimension; Hausdorff measure; conformal measures
UR - http://eudml.org/doc/216015
ER -

References

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  1. [1] K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. Zbl0587.28004
  2. [2] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979), 213-237. Zbl0422.28015
  3. [3] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, ibid. 38 (1981), 107-115. Zbl0456.28006
  4. [4] F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986), 359-386. Zbl0578.60069
  5. [5] F. Hofbauer, Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval, Studia Math. 103 (1992), 191-206. Zbl0813.28008
  6. [6] F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319-337. Zbl0702.58034
  7. [7] P. Raith, Hausdorff dimension for piecewise monotonic maps, Studia Math. 94 (1989), 17-33. Zbl0687.58013
  8. [8] P. Walters, An Introduction to Ergodic Theory, Springer, Berlin 1982. 

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