Topological Entropy and Adding Machine Maps
نویسندگان
چکیده
We prove two theorems which extend known results concerning periodic orbits and topological entropy in one-dimensional dynamics. One of these results concerns the adding machine map (also called the odometer map) fα defined on the α-adic adding machine ∆α. We let H(fα) denote the greatest lower bound of the topological entropies of F , taken over all continuous maps F of the interval which contain a copy of fα. We prove that if α is a sequence of primes such that 2 appears in the sequence exactly k times, then H(fα) = log 2 2k+1 .
منابع مشابه
Entropy of a semigroup of maps from a set-valued view
In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...
متن کاملEntropy Estimate for Maps on Forests
A 1993 result of J. Llibre, and M. Misiurewicz, (Theorem A [5]), states that if a continuous map f of a graph into itself has an s-horseshoe, then the topological entropy of f is greater than or equal to logs, that is h( f ) ? logs. Also a 1980 result of L.S. Block, J. Guckenheimer, M. Misiurewicz and L.S. Young (Lemma 1.5 [3]) states that if G is an A-graph of f then h(G) ? h( f ). In this pap...
متن کاملAn Ergodic Adding Machine on the Cantor Set
We calculate all ergodic measures for a specific function F on the unit interval. The supports of these measures consist of periodic orbits of period 2n and the classical ternary Cantor set. On the Cantor set, F is topologically conjugate to an “adding machine” in base 2. We show that F is representative of the class of functions with zero topological entropy on the unit interval, already analy...
متن کاملALGEBRAIC Zd-ACTIONS OF ENTROPY RANK ONE
We investigate algebraic Z-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of ...
متن کاملOn the Topological Entropy of Some Skew-Product Maps
The aim of this short note is to compute the topological entropy for a family of skew-product maps, whose base is a subshift of finite type, and the fiber maps are homeomorphisms defined in one dimensional spaces. We show that the skew-product map does not increase the topological entropy of the subshift.
متن کامل