# American Institute of Mathematical Sciences

January  2010, 26(1): 291-302. doi: 10.3934/dcds.2010.26.291

## Measure of full dimension for some nonconformal repellers

 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, Brazil

Received  January 2009 Revised  June 2009 Published  October 2009

Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of

$\frac{h_{\mu\circ \pi^{-1}}(S)}{\int \psi\circ\pi\d\mu}+ \frac{h_\mu(T)-h_{\mu\circ \pi^{-1}}(S)}{\int \varphi\d\mu}$

over all $T$-invariant Borel probability measures $\mu$ on $X$ is attained on the subset of ergodic measures. Here $h_\mu(T)$ stands for the metric entropy of $T$ with respect to $\mu$. As an application, we prove the existence of an ergodic invariant measure with full dimension for a class of transformations treated in [11], and also for the transformations treated in [17], where the author considers nonlinear skew-product perturbations of general Sierpinski carpets. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.

Citation: Nuno Luzia. Measure of full dimension for some nonconformal repellers. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 291-302. doi: 10.3934/dcds.2010.26.291
 [1] H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [2] Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367 [3] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [4] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [5] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [6] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [7] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [8] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [9] Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735 [10] Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098 [11] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 [12] Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125 [13] Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 [14] Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417 [15] Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 [16] Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006 [17] Cristina Lizana, Leonardo Mora. Lower bounds for the Hausdorff dimension of the geometric Lorenz attractor: The homoclinic case. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 699-709. doi: 10.3934/dcds.2008.22.699 [18] Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077 [19] Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 [20] Manuel Fernández-Martínez, Miguel Ángel López Guerrero. Generating pre-fractals to approach real IFS-attractors with a fixed Hausdorff dimension. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1129-1137. doi: 10.3934/dcdss.2015.8.1129

2019 Impact Factor: 1.338