# American Institute of Mathematical Sciences

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

## On dimensions of conformal repellers. Randomness and parameter dependency

 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.
Citation: Hans Henrik Rugh. On dimensions of conformal repellers. Randomness and parameter dependency. Discrete and 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. [2] R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ., 50 (1979), 259-273. [3] G. Birkhoff, "Lattice Theory," 3rd edition, American Mathematical Society Colloquium Publications, Vol. XXV, Amer. Math. Soc., Providence, RI, 1967. [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. [5] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [6] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Erg. Th. & Dyn. Syst., 17 (1997), 147-167. [7] C. Liverani, Decay of correlations, Annals Math. (2), 142 (1995), 239-301. doi: 10.2307/2118636. [8] D. Ruelle, Repellers for real analytic maps, Erg. Th. & Dyn. Syst., 2 (1982), 99-107. [9] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134. [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. [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. [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.

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. [2] R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ., 50 (1979), 259-273. [3] G. Birkhoff, "Lattice Theory," 3rd edition, American Mathematical Society Colloquium Publications, Vol. XXV, Amer. Math. Soc., Providence, RI, 1967. [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. [5] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909. [6] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Erg. Th. & Dyn. Syst., 17 (1997), 147-167. [7] C. Liverani, Decay of correlations, Annals Math. (2), 142 (1995), 239-301. doi: 10.2307/2118636. [8] D. Ruelle, Repellers for real analytic maps, Erg. Th. & Dyn. Syst., 2 (1982), 99-107. [9] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134. [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. [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. [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.
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