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指数函数族非回归参数的Hausdorff维数
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Abstract:
本文主要探讨指数函数族中一类特殊参数——非回归参数的Hausdorff维数问题，这一研究是在邱维元关于指数函数族逃逸参数的Hausdorff维数为2的重要发现基础上进行的自然延伸与深化。本文旨在证明，在给定的一个固定区域内，指数函数族中非回归且非逃逸参数的集合具有的Hausdorff维数严格小于某个给定的正数。
This paper primarily delves into the Hausdorff dimension of a special class of parameters within the exponential family—the non-recurrent parameters. This investigation constitutes a natural extension and deepening of Qiu’s seminal finding that the Hausdorff dimension of escaping parameters of the exponential family is 2. The objective of this paper is to prove that, within a given fixed region, the Hausdorff dimension of the set of non-escaping non-recurrent parameters in the exponential function family is strictly less than a specified positive number.
| [1] | Qiu, W.Y. (1994) Hausdorff Dimension of Them-Set of Λ Exp(z). Acta Mathematica Sinica, 10, 362-368. https://doi.org/10.1007/bf02582032 |
| [2] | Devaney, R.L. (1984) Julia Sets and Bifurcation Diagrams for Exponential Maps. Bulletin of the American Mathematical Society, 11, 167-171. https://doi.org/10.1090/s0273-0979-1984-15253-4 |
| [3] | Förster, M. and Schleicher, D. (2009) Parameter Rays in the Space of Exponential Maps. Ergodic Theory and Dynamical Systems, 29, 515-544. https://doi.org/10.1017/s0143385708000394 |
| [4] | Förster, M., Rempe, L. and Schleicher, D. (2007) Classification of Escaping Exponential Maps. Proceedings of the American Mathematical Society, 136, 651-663. https://doi.org/10.1090/s0002-9939-07-09073-9 |
| [5] | Bailesteanu, M., Balan, H.V. and Schleicher, D. (2007) Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints. Nonlinearity, 21, 113-120. https://doi.org/10.1088/0951-7715/21/1/006 |
| [6] | Tian, T. (2011) Parameter Rays for the Consine Family. Journal of Fudan University (Natural Science), 50, 10-22. |
| [7] | Huang, X. and Qiu, W. (2020) The Dimension Paradox in Parameter Space of Cosine Family. Chinese Annals of Mathematics, Series B, 41, 645-656. https://doi.org/10.1007/s11401-020-0223-8 |
| [8] | Cui, W.W. and Wang, J. (2023) Ergodic Exponential Maps with Escaping Singular Behaviours. https://doi.org/10.48550/arXiv.2308.10565 |
| [9] | Badeńska, A. (2010) Misiurewicz Parameters in the Exponential Family. Mathematische Zeitschrift, 268, 291-303. https://doi.org/10.1007/s00209-010-0671-z |
| [10] | Aspenberg, M. and Cui, W. (2024) Perturbations of Exponential Maps: Non-Recurrent Dynamics. Journal d’Analyse Mathématique. https://doi.org/10.1007/s11854-024-0340-5 |
