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Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent
On the uniqueness of an ergodic measure of full dimension for non-conformal repellers
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro, 21941-909, RJ, Brazil |
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.
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. |
[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. |
[4] |
R. Bowen,
Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-25.
|
[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. |
[6] |
M. Denker and M. Gordin,
Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.
doi: 10.1006/aima.1999.1843. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
N. Luzia,
Hausdorff dimension of certain random self-affine fractals, Stoch. Dyn., 11 (2011), 627-642.
doi: 10.1142/S0219493711003516. |
[14] |
C. McMullen,
The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J., 96 (1984), 1-9.
doi: 10.1017/S0027763000021085. |
[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. |
[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. |
[17] |
D. Ruelle,
Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107.
doi: 10.1017/S0143385700009603. |
[18] |
D. Ruelle,
Thermodynamic Formalism, 2$^{nd}$ edition, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511617546. |
[19] |
Ya. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation: Russian Math. Surveys, 27 (1972), 21-69. |
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. |
[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. |
[4] |
R. Bowen,
Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-25.
|
[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. |
[6] |
M. Denker and M. Gordin,
Gibbs measures for fibred systems, Adv. Math., 148 (1999), 161-192.
doi: 10.1006/aima.1999.1843. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
N. Luzia,
Hausdorff dimension of certain random self-affine fractals, Stoch. Dyn., 11 (2011), 627-642.
doi: 10.1142/S0219493711003516. |
[14] |
C. McMullen,
The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J., 96 (1984), 1-9.
doi: 10.1017/S0027763000021085. |
[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. |
[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. |
[17] |
D. Ruelle,
Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107.
doi: 10.1017/S0143385700009603. |
[18] |
D. Ruelle,
Thermodynamic Formalism, 2$^{nd}$ edition, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511617546. |
[19] |
Ya. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation: Russian Math. Surveys, 27 (1972), 21-69. |
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