For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$. The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $.

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

A mechanism that permits the separation and concentration of particles in natural systems is proposed. According to the mechanism, the particles in an aquifer are moved by an alternating flow of ground water. The retardation factor oscillates in synchronism with the alternating water flow. The retardation factor or the flow depend on the position in the aquifer (the particle velocity is u=v/R, where v is the water velocity and R is the retardation factor). The alternating flow is induced by tides, geysers or periodic rains. The alternating water flow produces temperature and water composition oscillations in the aquifer. Moreover, the alternating flow is caused by pressure oscillations: therefore, the retardation factor (which depends on pressure, temperature and water composition) oscillates. The dependence of the retardation factor on position is due to the inhomogeneity of the aquifer medium or to the position dependence of temperature, composition or pressure. Different particles may be concentrated at different places in the aquifer or in different aquifers. The particle concentration and the time necessary for concentration are estimated. The particles may be atoms, molecules, colloids or microorganisms. Such a mechanism may have played a role in chemical evolution.