A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks.
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| Title: | A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks. |
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| Authors: | Marzougui, Habib1 (AUTHOR) habib.marzougui@fsb.rnu.tn |
| Source: | Dynamical Systems: An International Journal. Sep2025, Vol. 40 Issue 3, p447-454. 8p. |
| Subjects: | Invariant measures, Diffeomorphisms, Dynamical systems, Dimension theory (Topology), Homeomorphisms, Functions of bounded variation, Ergodic theory, Critical point theory |
| Abstract: | Let f be a class P-homeomorphism of the circle $ S^{1} $ S 1 with several break points singularities, that are differentiable maps except at some singular points where the derivative has a jump, and such that $ \log \rm{Df} $ log Df has bounded variation on $ S^{1} $ S 1 . Assume that f has an irrational rotation number α of bounded density $ d_{\alpha } $ d α . Then the Hausdorff dimension $ \mathrm {dim}_{H}(\mu _{f}) $ dim H (μ f) of its unique f-invariant measure $ \mu _{f} $ μ f is bounded away from zero, i.e. such that $ \mathrm {dim}_{H}(\mu _{f})\geq C $ dim H (μ f) ≥ C , for some constant C>0 (depending only on $ d_{\alpha } $ d α and the total variation of $ \log \rm{Df} $ log Df ). This result generalizes and improves with a simple proof some results of Dzhalilov (Piecewise smoothness of conjugate homeomorphisms of a circle with corners, Theor. Math. Phys. 120 (1999), pp. 961–972) and Khanin and Kocić (Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point, Ergodic Theory Dyn. Syst. 39 (2017), pp. 1–9). [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | Let f be a class P-homeomorphism of the circle $ S^{1} $ S 1 with several break points singularities, that are differentiable maps except at some singular points where the derivative has a jump, and such that $ \log \rm{Df} $ log Df has bounded variation on $ S^{1} $ S 1 . Assume that f has an irrational rotation number α of bounded density $ d_{\alpha } $ d α . Then the Hausdorff dimension $ \mathrm {dim}_{H}(\mu _{f}) $ dim H (μ f) of its unique f-invariant measure $ \mu _{f} $ μ f is bounded away from zero, i.e. such that $ \mathrm {dim}_{H}(\mu _{f})\geq C $ dim H (μ f) ≥ C , for some constant C>0 (depending only on $ d_{\alpha } $ d α and the total variation of $ \log \rm{Df} $ log Df ). This result generalizes and improves with a simple proof some results of Dzhalilov (Piecewise smoothness of conjugate homeomorphisms of a circle with corners, Theor. Math. Phys. 120 (1999), pp. 961–972) and Khanin and Kocić (Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point, Ergodic Theory Dyn. Syst. 39 (2017), pp. 1–9). [ABSTRACT FROM AUTHOR] |
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| ISSN: | 14689367 |
| DOI: | 10.1080/14689367.2025.2472011 |
