The dimension spectrum of some dynamical systems. (English) Zbl 0683.58023
Summary: We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber. Using large-deviation theory, we prove, for some invariant measures of expanding Markov maps, that the Hausdorff dimension f(\(\alpha)\) of the set on which the measure has a singularity \(\alpha\) is a well-defined, concave, and regular function. In particular, we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number. We also show in these particular cases that the function f is universal.
MSC:
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |
Keywords:
Hausdorff dimension spectrum; partition function; critical circle map; universality; period doublingReferences:
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