Non-uniqueness and exotic solutions of conjugacy equations

Abstract

Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. It is well known that when conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties: they may be singular, fractal or even everywhere discontinuous. We provide an explicit formula showing how, in certain cases, a solution can be constructively determined. We find, for the same equation, remarkably distinct solutions.

Keywords::

• conjugacy equation
• functional equation
• explicit solution
• iterated function system

Keywords::

• 39B05
• 39B72
• 39B12
• 26A30

Acknowledgements

CS gratefully acknowledges the fruitful discussions with Prof. Fernando Ferreira regarding the conjunctive normal form. The authors acknowledge the meaningful and helpful comments of one anonymous referee.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author acknowledges support from Fundação para a Ciência e Tecnologia [grant number SFRH/BD/77623/2011]. The second author was partially supported by FCT [grant number UID/MAT/04561/2013].