# zbMATH — the first resource for mathematics

Local dimension for piecewise monotonic maps on the interval. (English) Zbl 0842.58019
The author considers piecewise monotonic maps of the interval and examines the pointwise dimension of measures. For a measure $$m$$, the pointwise dimension $$l_m(x)$$ at a point $$x$$ is given by the expression $$\lim_{r\downarrow 0} \log m(B_r(x))/\log r$$ where this exists. This quantity is closely related to the Hausdorff dimension of a measure and in particular, if $$l_m(x)$$ is constant almost everywhere with respect to $$m$$ then the Hausdorff dimension of $$m$$ is this constant value. The author shows that for piecewise monotonic maps which have derivatives of bounded $$p$$-variation, the pointwise dimension of an ergodic invariant measure with positive Lyapunov exponent $$\chi_m = \int \log |T'|dm$$ is given by $$h_m/\chi_m$$ almost everywhere, where $$h_m$$ is the entropy of the map with respect to $$m$$. He goes on to discuss the multifractal properties of conformal measures of expanding maps. Defining $$L_m(a)$$ to be $$\{x : l_m(x) = a\}$$, he considers the Hausdorff dimension of the sets $$L_m(a)$$. He shows that under some basic conditions, the quantity $$HD(L_m(a))$$ is a concave function of $$a$$ for $$a$$ in a closed bounded interval, this function arising as the Legendre transform of a pressure functional. In the course of the proof, the author extends the notion of topological pressure to deal with a larger class of functions than the traditional class of continuous functions.
Reviewer: A.Quas (Cambridge)

##### MSC:
 37E99 Low-dimensional dynamical systems 37A99 Ergodic theory 54H20 Topological dynamics (MSC2010) 28A78 Hausdorff and packing measures
Full Text:
##### References:
 [1] Denker, Ergodic Theory on Compact Spaces (1976) · Zbl 0328.28008 [2] Halsey, Phys. Rev. pp 1141– (1986) · Zbl 1184.37028 [3] DOI: 10.1017/S014338570000568X · Zbl 0691.58023 [4] Jones, Dynamics Reported, (vol. 4) pp 1– (1995) [5] Walters, An Introduction to Ergodic Theory. (1982) · Zbl 0475.28009 [6] Walters, Amer. J. Math. 97 pp 931– (1976) [7] DOI: 10.1017/S0143385700005162 · Zbl 0664.58022 [8] Raith, Studia Math. 94 pp 17– (1989) [9] Parry, Trans. Amer. Math. Soc. 112 pp 55– (1964) [10] Horn, Matrix Analysis. (1985) · Zbl 0576.15001 [11] DOI: 10.2307/2154735 · Zbl 0827.58036 [12] DOI: 10.1017/S014338570000955X · Zbl 0499.28012 [13] Hofbauer, Canadian Math. Bull. 35 pp 1– (1991) [14] DOI: 10.1007/BF01295288 · Zbl 0669.54021 [15] DOI: 10.1007/BF00334191 · Zbl 0578.60069 [16] Cutler, Nonlinear Time Series and Chaos, Vol. I: Dimension Estimation and Models. (1993)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.