| Abstract [eng] |
This thesis investigates the measure-theoretic behavior of a one-dimensional dynamical system \( ([0, 1), \mathcal{B}, m, \mathcal{S}) \), defined by the map \( \mathcal{S}(\xi) = \{\rho \xi\} \), where \( \rho > 1 \) is an irrational number and \( \{\cdot\} \) denotes the fractional part. The study focuses on three specific cases: \( \mathcal{S}_1(\xi) = \{G \xi\} \), with \( G = \frac{1+\sqrt{5}}{2} \) (the golden ratio); \( \mathcal{S}_2(\xi) = \{\rho_2 \xi\} \), with \( \rho_2 = \frac{1+\sqrt{3}}{2} \); and \( \mathcal{S}_3(\xi) = \{\rho_3 \xi\} \), with \( \rho_3 = \frac{1+\sqrt{7}}{2} \). For each transformation, the associated Perron operator is derived explicitly, revealing a piecewise structure due to a discontinuity at \( \xi = \frac{1}{\rho} \). It is shown analytically that the Lebesgue measure is not invariant under any of these maps. For the golden ratio map, a known piecewise constant invariant density is revisited and verified. In the cases of \( \rho_2 \) and \( \rho_3 \), analytical derivation of invariant densities proved intractable, and instead, numerical estimation was employed. Empirical frequency histograms generated from long orbit simulations (20000 iterations) provided approximate invariant densities for these maps, capturing their non-uniform statistical structure. The results demonstrate that even simple linear transformations with irrational scaling can exhibit complex invariant behavior, and that measure preservation is not implied by linearity or irrationality alone. The Perron framework effectively captures the redistribution of mass over time and highlights the structural influence of the map’s discontinuity on invariant measures. |
