Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for anarbitrarydynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—thet-entropy.

On Symbolic Dynamics of One-Humped Maps of the Interval

We present an explicit construction of minimal deterministic automata which accept the languages of L-R symbolic sequences of unimodal maps resp. arbitrarily close approximations thereof. They are used to study a recently introduced complexity measure of this language which we conjecture to be a new invariant under diffeomorphisms. On each graph corresponding to such an automaton, the evolution is a topological Markov chain which does not seem to correspond to a partition of the interval into a countable number of intervals.

On the subsystems of topological Markov chains

Let SA be an irreducible and aperiodic topological Markov chain. If SĀ is an irreducible and aperiodic topological Markov chain, whose topological entropy is less than that of SA, then there exists an irreducible and aperiodic topological Markov chain, whose topological entropy equals the topological entropy at SĀ, and that is a subsystem of SA. If S is an expansive homeomorphism of the Cantor discontinuum, whose topological entropy is less than that of SA, and such that for every j∈ℕ the number of periodic points of least period j of S is less than or equal to the number of periodic points of least period j of SA, then S is topological conjugate to a subsystem of SA.