Matching for parameterised symmetric golden maps

Abstract: In 2020, Dajani and Kalle investigated invariant measures and frequencies of digits of signed binary expansions arising from a parameterised family of piecewise linear interval maps of constant slope 2. Central to their study was a property called ‘matching,’ where the orbits of the left and right limits of discontinuity points agree after some finite number of steps. We obtain analogous results for a parameterised family of ‘symmetric golden maps’ of constant slope $\beta$, with $\beta$ the golden mean. Matching is again central to our methods, though the dynamics of the symmetric golden maps are more delicate than the binary case. We characterize the matching phenomenon in our setting, present explicit invariant measures and frequencies of digits of signed $\beta$-expansions, and---time permitting---show further implications for a family of piecewise linear maps which arise as jump transformations of the symmetric golden maps.

Joint with Karma Dajani.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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