Measures of full dimension on affine-invariant sets

Abstract We determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.

[1]  P. Billingsley,et al.  Ergodic theory and information , 1966 .

[2]  V. Rokhlin.  LECTURES ON THE ENTROPY THEORY OF MEASURE-PRESERVING TRANSFORMATIONS , 1967 .

[3]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[4]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[5]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.

[6]  F. Ledrappier,et al.  The metric entropy of diffeomorphisms , 1984 .

[7]  Curtis T. McMullen,et al.  The Hausdorff dimension of general Sierpiński carpets , 1984, Nagoya Mathematical Journal.

[8]  F. Ledrappier,et al.  The metric entropy of diffeomorphisms Part II: Relations between entropy, exponents and dimension , 1985 .

[9]  Tim Bedford,et al.  Generating special Markov partitions for hyperbolic toral automorphisms using fractals , 1986, Ergodic Theory and Dynamical Systems.

[10]  T. Cover,et al.  A sandwich proof of the Shannon-McMillan-Breiman theorem , 1988 .

[11]  R. Daniel Mauldin,et al.  Hausdorff dimension in graph directed constructions , 1988 .

[12]  T. Bedford.  On Weierstrass-like functions and random recurrent sets , 1989, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  T. Bedford.  The box dimension of self-affine graphs and repellers , 1989 .

[14]  A Shannon-McMillan theorem for motley names , 1990 .

[15]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[16]  Yuval Peres,et al.  Intersecting random translates of invariant Cantor sets , 1991 .

[17]  Steven P. Lalley,et al.  Hausdorff and box dimensions of certain self-affine fractals , 1992 .

[18]  Kenneth Falconer,et al.  The dimension of self-affine fractals II , 1992, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  F. Ledrappier.  On the dimension of some graphs , 1992 .

[20]  Yuval Peres,et al.  Hausdorff dimensions of sofic affine-invariant sets , 1996 .