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Dimensions

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Encyclopedia of Nonlinear Science

DIMENSIONS

The classical, integer-valued definition of dimension (see Hurewicz & Wallman, 1941) is defined inductively: the empty set has dimension −1, and a set has dimension n if n is the smallest integer such that every point has arbitrarily small neighborhoods whose boundaries have dimension less than n. This gives the “right” answer for smooth curves and surfaces, whose dimension we know intuitively.

In order to describe more accurately the complicated fractal sets that arise in nonlinear dynamics, we need to introduce more subtle definitions. Surprisingly, there are several generalizations of dimensions that still assign the intuitively correct dimensions to the above-noted well-behaved sets, and we recall two of them here.

The first of these is the “box-counting” dimension, also known as the Minkowski dimension, the fractal dimension, the entropy dimension, the capacity dimension, and the limit capacity: a litany of names that testifies to its popularity. For a subset X of take a fixed array of boxes of side δ, and count the number N_δ(X) of these boxes that intersect with X. If N_δ(X)~δ^−d as δ→0, then X has box-counting dimension d. This can be made mathematically precise by defining

(1)

(For alternative definitions that give the same quantity see Falconer (1990).) While the box-counting dimension is simple to define, it is not without problems. For example, the set an unlikely candidate for a fractal, has box-counting dimension We now introduce another widely used definition of dimension that does not suffer from this anomaly.

We could try to define a notion of the “d-dimensional volume” of X as the limit of N_δ(X)δ^d as δ→0, but such a definition does not even agree with the standard definition of volume (Lebesgue measure) when d is an integer. Instead, the proper generalization of Lebesgue measure to non-integer dimensions is d-dimensional Hausdorff measure. Essentially, we cover a set by a collection of balls of radii r_i≤δ, and then let be the limit of as δ tends to zero. More precisely, we define

(the notation B_r(x) denotes an open ball centered at x of radius r). The resulting measure is proportional to Lebesgue measure when d is an integer. The “Hausdorff dimension” of X is the smallest value of d for which is finite,

Since µ_d(X, δ)≤N_δ(X)δ^d, we always have d_H(X)≤ d_box(X) (and this inequality can be strict: the set S defined above has zero Hausdorff dimension). While harder to estimate in practice, the Hausdorff dimension is easier to deal with theoretically.

If we want to estimate the dimension of the attractor of a dynamical system, it is useful to have a method based on dynamical quantities. In 1980, Douady & Oesterlé showed how to obtain a bound on the dimension of the attractor of an iterated C¹ map f on Denote by D f(x) the matrix of partial derivatives of f, i.e., [D f]_ij=∂f_i/∂x_j, and let λ₁(x)≥λ₂(x)≥λ_n(x) be the logarithms of the eigenvalues of [D f(x)^TD f(x)]^1/2. Now set

(2)

where j is the largest integer for which λ₁(x)+ …+λ_j(x)≥0 (note that j≤d(x)<j+1). If d> d(x), then any infinitesimal d-volume near x is contracted under the application of f, so

(3)

Hunt (1996) showed that the right-hand side of (3) also bounds (A similar approach also works for the attractors of flows by taking f to be the time T map, for some suitable T. Constantin & Foias (1985) have proved a version of (3) for the attractors of infinite-dimensional dynamical systems.)

However, the box-counting and Hausdorff dimensions give equal weighting to all points in the attractor, while it is possible to have regions of the attractor that are visited very rarely. In such a situation, it can be more natural to consider invariant measures rather than attractors. As a (canonical) example of such a measure, suppose that generates a dynamical system on Then for any set X, we can define

where x is a point in the basin of attraction of The quantity μ(X) is the proportion of time spent in X by a “typical trajectory” on the attractor.

There are various ways of defining the dimension of a measure µ. We could define the Hausdorff/box-counting dimension of μ to be the dimension of its support,

d_box/H(μ)=inf{d_box/H(E):μ(E)=1},

but this still discounts the dynamical information contained in µ. Kaplan & Yorke (1979) defined the Lyapunov dimension of μ, d_L(μ), precisely as in (2), but replacing λ_j(x) by the Lyapunov exponents associated with μ (the asymptotic growth rates of infinitesimal displacements about trajectories through µ-almost every choice of initial condition). In 1981, Ledrappier showed that for a very general class of dynamical systems, d_H(µ)≤d_L(µ) (the inequality can be strict), while

(Kaplan & Yorke had originally conjectured that

We now give two definitions of dimension that take into account the spatial structure of µ. The correlation at scale δ is defined by

which gives the probability that two points chosen according to the probability measure μ lie within δ of each other. If C(δ)~δ^d as δ→0, then d is the correlation dimension d_corr(μ). This was introduced by Grassberger & Procaccia (1983), who demonstrated that this quantity is particularly suited to numerical calculation.

Alternatively, define the “δ-entropy” K_δ(µ)= −∑_i μ(B_i) ln µ(B_i), where {B_i} is an array of boxes of side δ; the information dimension is given by

(Ruelle (1989) refers to d_H(µ) as the “information dimension” of µ: this should serve to emphasize how important it is when discussing dimensions to be explicit about the definition.)

Three of these dimensions occur as part of a scale of dimension-like quantities (see Grassberger, 1983). If B_i is an array of boxes of side δ, set

(“the Renyi q entropy”). Note that K_δ(0)=log N_δ (supp µ),

and that since we have log Now define the Renyi dimensions D_q(µ) by

Then D₀(µ)=d_box(µ), D₁(µ)=d_inf(µ), and D₂(µ)= d_corr(μ). Since D_q is non-increasing in q, we have in particular d_corr(μ)≤d_inf(µ)≤d_box(µ).

The Renyi dimensions are similar to quantities used to define the “multi-fractal spectrum.” The theory relates the numbers τ(q)=(1−q)D_q to the frequency of various scaling behaviors about points on a fractal set: roughly, if for some small ε the number of δ-mesh cubes B_i with

δ^α+ε≤μ(B_i)<δ^α

scales like δ^−f(α), then

f(α(q))=τ(q)+qα(q),

where q=f′(α(q)). The curve f(α) is the multi-fractal spectrum of the measure µ. (As remarked by Falconer (1990), the tempting interpretation of the “fractal spectrum” as the dimension of sets of points x where μ(B_δ(x))~δ^α is incorrect: typically the dimension of such sets will be zero or the same as that of the whole space, see Genyuk (1997/98).) Although these ideas have proved useful in the theory of turbulence (e.g. Frisch, 1995), their mathematical foundations have still to be fully resolved.

JAMES C.ROBINSON

Dimensions

DIMENSIONS

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