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On the uniqueness of an ergodic measure of full dimension for non-conformal repellers
Non-degenerate locally connected models for plane continua and Julia sets
1. | Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA |
2. | Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva St., 119048 Moscow, Russia |
Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of subcontinua with certain properties. Applications to complex polynomial dynamics are discussed.
References:
[1] |
A. Blokh and L. Oversteegen,
Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc., 356 (2004), 119-133.
doi: 10.1090/S0002-9947-03-03415-9. |
[2] |
A. Blokh, C. Curry and L. Oversteegen,
Locally connected models for Julia sets, Advances in Math, 226 (2011), 1621-1661.
doi: 10.1016/j.aim.2010.08.011. |
[3] |
A. Blokh, C. Curry and L. Oversteegen,
Finitely Suslinian models for planar compacta with applications to Julia sets, Proc. Amer. Math. Soc., 141 (2013), 1437-1449.
doi: 10.1090/S0002-9939-2012-11607-7. |
[4] |
A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin,
Quadratic-like dynamics of cubic polynomials, Communications in Mathematical Physics, 341 (2016), 733-749.
doi: 10.1007/s00220-015-2559-6. |
[5] |
A. Blokh and L. Oversteegen,
Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
|
[6] |
B. Branner and J. Hubbard,
The iteration of cubic polynomials, Part Ⅰ: The global topology of parameter space, Acta Math., 160 (1988), 143-206.
doi: 10.1007/BF02392275. |
[7] |
H. Cremer,
Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163.
doi: 10.1007/BF01451586. |
[8] |
A. Douady and J. H. Hubbard,
On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup.(4), 18 (1985), 287-343.
doi: 10.24033/asens.1491. |
[9] |
J. Kiwi,
$\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267.
doi: 10.1016/S0001-8708(03)00144-0. |
[10] |
K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608. |
[11] |
J. Milnor,
Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp. |
show all references
References:
[1] |
A. Blokh and L. Oversteegen,
Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc., 356 (2004), 119-133.
doi: 10.1090/S0002-9947-03-03415-9. |
[2] |
A. Blokh, C. Curry and L. Oversteegen,
Locally connected models for Julia sets, Advances in Math, 226 (2011), 1621-1661.
doi: 10.1016/j.aim.2010.08.011. |
[3] |
A. Blokh, C. Curry and L. Oversteegen,
Finitely Suslinian models for planar compacta with applications to Julia sets, Proc. Amer. Math. Soc., 141 (2013), 1437-1449.
doi: 10.1090/S0002-9939-2012-11607-7. |
[4] |
A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin,
Quadratic-like dynamics of cubic polynomials, Communications in Mathematical Physics, 341 (2016), 733-749.
doi: 10.1007/s00220-015-2559-6. |
[5] |
A. Blokh and L. Oversteegen,
Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
|
[6] |
B. Branner and J. Hubbard,
The iteration of cubic polynomials, Part Ⅰ: The global topology of parameter space, Acta Math., 160 (1988), 143-206.
doi: 10.1007/BF02392275. |
[7] |
H. Cremer,
Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163.
doi: 10.1007/BF01451586. |
[8] |
A. Douady and J. H. Hubbard,
On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup.(4), 18 (1985), 287-343.
doi: 10.24033/asens.1491. |
[9] |
J. Kiwi,
$\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267.
doi: 10.1016/S0001-8708(03)00144-0. |
[10] |
K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608. |
[11] |
J. Milnor,
Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp. |
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