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November  2017, 37(11): 5763-5780. doi: 10.3934/dcds.2017250

On the uniqueness of an ergodic measure of full dimension for non-conformal repellers

Nuno Luzia

Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro, 21941-909, RJ, Brazil

Received  June 2016 Revised  June 2017 Published  July 2017

We give a subclass $\mathcal{L}$ of Non-linear Lalley-Gatzouras carpets and an open set $\mathcal{U}$ in $\mathcal{L}$ such that any carpet in $\mathcal{U}$ has a unique ergodic measure of full dimension. In particular, any Lalley-Gatzouras carpet which is close to a non-trivial general Sierpinski carpet has a unique ergodic measure of full dimension.

Keywords: Measure of full dimension, non-conformal repeller.
Mathematics Subject Classification: Primary: 37D35, 37C45.
Citation: Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250
References:
[1]

J. Barral and D.-J. Feng, Non-uniqueness of ergodic measures with full Hausdorff dimension on Gatzouras-Lalley carpet, Nonlinearity, 24 (2011), 2563-2567.  doi: 10.1088/0951-7715/24/9/010.  Google Scholar

[2]

T. Bedford, Crinkly Curves, Markov Partitions and Box Dimension of Self Similar Sets, Ph. D thesis, University of Warwick, 1984. Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, 1975.  Google Scholar

[4]

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-25.   Google Scholar

[5]

T. Das and D. Simmons, The Hausdorff and dynamical dimensions of self-affine sponges: A dimension gap result, Invent. Math., (2017), 1-50.  doi: 10.1007/s00222-017-0725-5.  Google Scholar

[6]

M. Denker and M. Gordin, Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.  doi: 10.1006/aima.1999.1843.  Google Scholar

[7]

M. Denker, M. Gordin and S. Heinemann, On the relative variational principle for fibred expanding maps, Ergod. Th. & Dynam. Sys., 22 (2002), 757-782.  doi: 10.1017/S014338570200038X.  Google Scholar

[8]

D.-J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.  Google Scholar

[9]

D. Gatzouras and P. Lalley, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J., 41 (1992), 533-568.  doi: 10.1512/iumj.1992.41.41031.  Google Scholar

[10]

R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. Th. & Dynam. Sys., 16 (1996), 307-323.  doi: 10.1017/S0143385700008828.  Google Scholar

[11]

N. Luzia, A variational principle for the dimension for a class of non-conformal repellers, Ergod. Th. & Dynam. Sys., 26 (2006), 821-845.  doi: 10.1017/S0143385705000659.  Google Scholar

[12]

N. Luzia, Measure of full dimension for a class of nonconformal repellers, Discrete Contin. Dyn. Syst., 26 (2010), 291-302.  doi: 10.3934/dcds.2010.26.291.  Google Scholar

[13]

N. Luzia, Hausdorff dimension of certain random self-affine fractals, Stoch. Dyn., 11 (2011), 627-642.  doi: 10.1142/S0219493711003516.  Google Scholar

[14]

C. McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J., 96 (1984), 1-9.  doi: 10.1017/S0027763000021085.  Google Scholar

[15]

E. Olivier, Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus, Ergod. Th. & Dynam. Sys., 30 (2010), 1503-1528.  doi: 10.1017/S0143385709000546.  Google Scholar

[16]

F. Przytycki and M. Urbanski, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.  Google Scholar

[17]

D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107.  doi: 10.1017/S0143385700009603.  Google Scholar

[18]

D. Ruelle, Thermodynamic Formalism, 2$^{nd}$ edition, Cambridge University Press, 2004. doi: 10.1017/CBO9780511617546.  Google Scholar

[19]

Ya. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation: Russian Math. Surveys, 27 (1972), 21-69.  Google Scholar

show all references

References:
[1]

J. Barral and D.-J. Feng, Non-uniqueness of ergodic measures with full Hausdorff dimension on Gatzouras-Lalley carpet, Nonlinearity, 24 (2011), 2563-2567.  doi: 10.1088/0951-7715/24/9/010.  Google Scholar

[2]

T. Bedford, Crinkly Curves, Markov Partitions and Box Dimension of Self Similar Sets, Ph. D thesis, University of Warwick, 1984. Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, 1975.  Google Scholar

[4]

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-25.   Google Scholar

[5]

T. Das and D. Simmons, The Hausdorff and dynamical dimensions of self-affine sponges: A dimension gap result, Invent. Math., (2017), 1-50.  doi: 10.1007/s00222-017-0725-5.  Google Scholar

[6]

M. Denker and M. Gordin, Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.  doi: 10.1006/aima.1999.1843.  Google Scholar

[7]

M. Denker, M. Gordin and S. Heinemann, On the relative variational principle for fibred expanding maps, Ergod. Th. & Dynam. Sys., 22 (2002), 757-782.  doi: 10.1017/S014338570200038X.  Google Scholar

[8]

D.-J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.  Google Scholar

[9]

D. Gatzouras and P. Lalley, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J., 41 (1992), 533-568.  doi: 10.1512/iumj.1992.41.41031.  Google Scholar

[10]

R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. Th. & Dynam. Sys., 16 (1996), 307-323.  doi: 10.1017/S0143385700008828.  Google Scholar

[11]

N. Luzia, A variational principle for the dimension for a class of non-conformal repellers, Ergod. Th. & Dynam. Sys., 26 (2006), 821-845.  doi: 10.1017/S0143385705000659.  Google Scholar

[12]

N. Luzia, Measure of full dimension for a class of nonconformal repellers, Discrete Contin. Dyn. Syst., 26 (2010), 291-302.  doi: 10.3934/dcds.2010.26.291.  Google Scholar

[13]

N. Luzia, Hausdorff dimension of certain random self-affine fractals, Stoch. Dyn., 11 (2011), 627-642.  doi: 10.1142/S0219493711003516.  Google Scholar

[14]

C. McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J., 96 (1984), 1-9.  doi: 10.1017/S0027763000021085.  Google Scholar

[15]

E. Olivier, Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus, Ergod. Th. & Dynam. Sys., 30 (2010), 1503-1528.  doi: 10.1017/S0143385709000546.  Google Scholar

[16]

F. Przytycki and M. Urbanski, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.  Google Scholar

[17]

D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107.  doi: 10.1017/S0143385700009603.  Google Scholar

[18]

D. Ruelle, Thermodynamic Formalism, 2$^{nd}$ edition, Cambridge University Press, 2004. doi: 10.1017/CBO9780511617546.  Google Scholar

[19]

Ya. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation: Russian Math. Surveys, 27 (1972), 21-69.  Google Scholar

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