APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

2011 ◽  
Vol 11 (04) ◽  
pp. 643-679 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Sierpinski Carpet ◽  
3 Dimensional ◽  
Sofic Shift ◽  
Full Shift ◽  
Sierpiński Carpet ◽  
Invariant Ergodic Measure

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

scholarly journals Dimensions of compact invariant sets of some expanding maps

2009 ◽  
Vol 29 (1) ◽  
pp. 281-315 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Invariant Set ◽  
Invariant Sets ◽  
Conformal Map ◽  
Expanding Maps ◽  
Sierpinski Carpet ◽  
Shift Of Finite Type ◽  
Unique Measure ◽  
Sierpiński Carpet

AbstractWe study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a ‘general Sierpiński carpet’ was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpiński carpet represented by a shift of finite type. We give some conditions under which a general Sierpiński carpet has a unique measure of full Hausdorff dimension and study the properties of the unique measure.

scholarly journals Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

2012 ◽  
Vol 33 (3) ◽  
pp. 831-850 ◽  
Author(s):  
YONGLUO CAO ◽  
HUYI HU ◽  
YUN ZHAO
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Topological Pressure ◽  
Compact Sets ◽  
Equivalent Definition ◽  
Ergodic Measures ◽  
Nonadditive Measure ◽  
Definition Of ◽  
Conformal Repeller

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

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