Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function ${{f\colon\mathbb{R}^n \to \mathbb{R}}}$ to be vertically rigid if graph(cf) is isometric to graph(f) for every c ≠?0. It is horizontally rigid if graph ${(f(c \vec{x}))}$ is isometric to graph(f) for every c ≠?0 (see Cain et?al., Real Anal. Exch. 31:515–518, 2005/2006). In Balka and Elekes (J. Math. Anal. Appl. 345:880–888, 2008) the authors of the present paper settled Jankovi?’s conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form a?+?bx or ${{a+be^{kx} (a,b,k \in \mathbb{R})}}$ . Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139–156, 2009) that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form ${a + bx + dy, a +s(y)e^{kx}}$ or ${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$ . The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343–354, 2009) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form ${{a+bx (a,b\in \mathbb{R})}}$ . The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form ${{a+ bx + dy (a,b,d \in \mathbb{R})}}$ . This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

The structure of rigid functions

A function f:R→R is called vertically rigid if graph(cf) is isometric to graph(f) for all c≠0. We prove Jankovi?'s conjecture by showing that a continuous function is vertically rigid if and only if it is of the form a+bx or a+bekx (a,b,k∈R). We answer the question of Cain, Clark and Rose by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case, consider functions bounded on some interval and functions with at least one point of continuity. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity.