By Luís Barreira

The measurement concept of dynamical platforms has steadily constructed, specifically during the last twenty years, into an self reliant and very lively box of study. Its major objective is to review the complexity of units and measures which are invariant less than the dynamics. particularly, it truly is necessary to characterizing chaotic unusual attractors. up to now, a few components of the idea have both merely been defined, simply because they are often lowered to the case of maps, or are too technical for a much wider viewers. during this recognize, the current monograph is meant to supply a finished advisor. furthermore, the textual content is self-contained and aside from a few easy ends up in Chapters three and four, the entire ends up in the publication comprise distinct proofs.

The e-book is meant for researchers and graduate scholars focusing on dynamical platforms who desire to have a sufficiently entire view of the idea including a operating wisdom of its major concepts. The dialogue of a few open difficulties can also be integrated within the desire that it may possibly bring about extra advancements. preferably, readers must have a few familiarity with the elemental notions and result of ergodic thought and hyperbolic dynamics on the point of an introductory path within the sector, although the preliminary chapters additionally evaluation all of the invaluable material.

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Chapter 6 Pointwise Dimension and Applications In this chapter, again for conformal hyperbolic flows, we establish an explicit formula for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and the Lyapunov exponents. In particular, this formula allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in each ergodic decomposition. We also discuss the problem of the existence of invariant measures of maximal dimension.

K}Z given by ΣA = (· · · i−1 i0 i1 · · · ) : ain in+1 = 1 for n ∈ Z , and the shift map σ : ΣA → ΣA defined by σ (· · · i0 · · · ) = (· · · j0 · · · ), where jn = in+1 for each n ∈ Z. 5 The map σ |ΣA is said to be a (two-sided) topological Markov chain with transition matrix A. 3 Symbolic Dynamics 37 We define a coding map π : ΣA → k i=1 Ri for the hyperbolic set by (T |Z)−j (int Rij ). π(· · · i0 · · · ) = j ∈Z One can easily verify that π ◦ σ = T ◦ π. 9) Given β > 1, we equip ΣA with the distance d given by ∞ d (· · · i−1 i0 i1 · · · ), (· · · j−1 j0 j1 · · · ) = β −|n| |in − jn |.

Given a set Z ⊂ Rm and α ∈ R, the α-dimensional Hausdorff measure of Z is defined by m(Z, α) = lim inf ε→0 U (diam U )α , U ∈U where the infimum is taken over all finite or countable collections U with diam U ≤ ε such that U ∈U U ⊃ Z. 6 The Hausdorff dimension of Z ⊂ Rm is defined by dimH Z = inf{α ∈ R : m(Z, α) = 0}. The lower and upper box dimensions of Z ⊂ Rm are defined respectively by dimB Z = lim inf ε→0 log N (Z, ε) − log ε and dimB Z = lim sup ε→0 log N (Z, ε) , − log ε where N(Z, ε) is the least number of balls of radius ε that are needed to cover the set Z.