Hyperbolic tilings and formal language theory

In [11], it was shown that a few languages constructed from some figures of hyperbolic tilings cannotbe recognized by pushdown automata but they can be recognized by a 2-iterated pushdown automaton.Before, it was known that several tessellations of the hyperbolic plane are generated by substitutions,see [3]. This property is also clear from [7].These substitutions can be also described by the use of grammars. This is rather straightforward. In[6], these substitutions appear as rules of a grammar, although the grammar is not formally described.Iterated pushdown automata were introduced in [4, 12] and we refer the reader to [1] for referencesand for the connection of this topic with sequences of rational numbers. By their definition, iteratedpushdown automata are more powerful than standard pushdown automata but they are far less powerfulthan Turing machines. As Turing machines can be simulated by a finite automaton with two independentstacks, iterated pushdown automata can be viewed as an intermediate device, see also [5] for otherconnections of automata with graph algebras.In this paper, we show an application of this device to the characterization of contour words ofa family of bounded domains in many tilings of the hyperbolic plane. We can do the same kind ofapplication for a tiling of the hyperbolic 3D space and for another one in the hyperbolic 4D space. Thesetwo latter applications cannot be generalized to any dimension as, starting from dimension 5, there is notiling of the hyperbolic space which would be a tessellation generated by a regular polytope.In Section 2, we remember the definition of iterated pushdown automata with an application to thecomputation of the recognition of words of the form a