The Michigan Mathematical Journal

The finer geometry and dynamics of the hyperbolic exponential family

Mariusz Urbański and Anna Zdunik

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Michigan Math. J., Volume 51, Issue 2 (2003), 227-250.

First available in Project Euclid: 4 August 2003

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Zentralblatt MATH identifier

Primary: 37F35: Conformal densities and Hausdorff dimension 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 37O35 37A05: Measure-preserving transformations


Urbański, Mariusz; Zdunik, Anna. The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J. 51 (2003), no. 2, 227--250. doi:10.1307/mmj/1060013195.

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  • K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37--60.
  • R. Bhattacharjee and R. Devaney, Tying hairs for structurally stable exponentials, Ergodic Theory Dynam. Systems 20 (2000), 1603--1617.
  • R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11--25.
  • M. Denker and M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), 563--587.
  • ------, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365--384.
  • ------, Geometric measures for parabolic rational maps, Ergodic Theory Dynam. Systems 12 (1992), 53--66.
  • H. F. Federer, Geometric measure theory, Springer-Verlag, New York, 1969.
  • F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller theory, Amer. Math. Soc., Providence, RI, 2000.
  • M. Guzmán, Differentiation of integrals in $\Bbb R^n,$ Measure theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Math., 541, pp. 181--185, Springer-Verlag, Berlin, 1976.
  • B. Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of $\lambdae^z$ and $\lambda\,\text\rm sin\,z,$ Fund. Math. 159 (1999), 269--287.
  • O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, Berlin, 1973.
  • M. Martens, The existence of $\sigma$-finite invariant measures: Applications to real one-dimensional dynamics, preprint.
  • C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329--342.
  • F. Przytycki and M. Urbański, Fractals in the plane---the ergodic theory methods, to appear [available at $\langle $$\sim$}urbanski$\rangle $].
  • S. J. Taylor and C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679--699.