For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of HeinonenKoskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.

The notion of topological entropy dimension for a Z-action has been introduced to measure the subexponential complexity of zero entropy systems. Given a Z2-action, along with a Z2-entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a Z2-action and the directional entropy dimensions of its subactions satisfy ce...

A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A ⊆ Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn can be constructed from A using finitely many boolea...

The border collision normal form is a continuous piecewise affine map of R n with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence that the attractors have Haus-dorff (fractal) dimension n. If n = 2 the attractors have topological dimension two, i.e. they contain open sets, and...

In this paper, we introduce and characterize the dynamically deened recurrence dimension, a new topological invariant number, following the idea of a previous article 9], but with some modiications and improvements. We study some example of Toeplitz subshifts for which we can show that the recurrence dimension is a topologically invariant number diierent from the topological entropy.

Topological manifolds form an important class of topological spaces with applications throughout mathematics. However, the development of this scienti c area, even at the initial stage, requires non-trivial Brouwer's theorems: the xed point theorem, the topological invariance of degree, and the topological invariance of dimension, where each of them is provided for n-dimensional case. We presen...