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July  2012, 32(7): 2553-2564. doi: 10.3934/dcds.2012.32.2553

On dimensions of conformal repellers. Randomness and parameter dependency

Hans Henrik Rugh 1,

1. 

Department of Mathematics, University of Paris Sud 11, F-91405 Orsay, France

Received  December 2009 Revised  July 2010 Published  March 2012

We consider random conformal repellers. We show how to apply Bowen's formula for the Hausdorff dimension in this context and prove smoothness of the dimension with respect to parameters. The present article is essentially an extract of [11]. Our aim here is to emphasize the ideas and mechanisms behind rather than mathematical rigor.
Keywords: Hausdorff dimension., Dimension theory, self-similarity, random conformal dynamics, repeller.
Mathematics Subject Classification: Primary: 37C45, 28A78; Secondary: 60D05, 37H1.
Citation: Hans Henrik Rugh. On dimensions of conformal repellers. Randomness and parameter dependency. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2553-2564. doi: 10.3934/dcds.2012.32.2553
References:
[1]

L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Erg. Th. & Dyn. Syst., 16 (1996), 871-927.  Google Scholar

[2]

R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ., 50 (1979), 259-273. Google Scholar

[3]

G. Birkhoff, "Lattice Theory," 3rd edition, American Mathematical Society Colloquium Publications, Vol. XXV, Amer. Math. Soc., Providence, RI, 1967.  Google Scholar

[4]

K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554. doi: 10.1090/S0002-9939-1989-0969315-8.  Google Scholar

[5]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.  Google Scholar

[6]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Erg. Th. & Dyn. Syst., 17 (1997), 147-167.  Google Scholar

[7]

C. Liverani, Decay of correlations, Annals Math. (2), 142 (1995), 239-301. doi: 10.2307/2118636.  Google Scholar

[8]

D. Ruelle, Repellers for real analytic maps, Erg. Th. & Dyn. Syst., 2 (1982), 99-107.  Google Scholar

[9]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134.  Google Scholar

[10]

H. H. Rugh, Coupled maps and analytic function spaces, Ann. Scient. Éc. Norm. Sup. (4), 35 (2002), 489-535. doi: 10.1016/S0012-9593(02)01102-3.  Google Scholar

[11]

H. H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Ann. Math. (2), 168 (2008), 695-748. doi: 10.4007/annals.2008.168.695.  Google Scholar

[12]

M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Erg. Th. & Dyn. Syst., 24 (2004), 279-315.  Google Scholar

show all references

References:
[1]

L. M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Erg. Th. & Dyn. Syst., 16 (1996), 871-927.  Google Scholar

[2]

R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ., 50 (1979), 259-273. Google Scholar

[3]

G. Birkhoff, "Lattice Theory," 3rd edition, American Mathematical Society Colloquium Publications, Vol. XXV, Amer. Math. Soc., Providence, RI, 1967.  Google Scholar

[4]

K. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc., 106 (1989), 543-554. doi: 10.1090/S0002-9939-1989-0969315-8.  Google Scholar

[5]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.  Google Scholar

[6]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Erg. Th. & Dyn. Syst., 17 (1997), 147-167.  Google Scholar

[7]

C. Liverani, Decay of correlations, Annals Math. (2), 142 (1995), 239-301. doi: 10.2307/2118636.  Google Scholar

[8]

D. Ruelle, Repellers for real analytic maps, Erg. Th. & Dyn. Syst., 2 (1982), 99-107.  Google Scholar

[9]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134.  Google Scholar

[10]

H. H. Rugh, Coupled maps and analytic function spaces, Ann. Scient. Éc. Norm. Sup. (4), 35 (2002), 489-535. doi: 10.1016/S0012-9593(02)01102-3.  Google Scholar

[11]

H. H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency, Ann. Math. (2), 168 (2008), 695-748. doi: 10.4007/annals.2008.168.695.  Google Scholar

[12]

M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Erg. Th. & Dyn. Syst., 24 (2004), 279-315.  Google Scholar

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