In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure seque...

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on underlying simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented.

The fast dynamo growth rate for a C k+1 map or ow is bounded above by topo-logical entropy plus a 1=k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following anti-dynamo theorem: in C 1 systems fast dynamo action is not possible without the presence of chaos. In addition topologica...

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down a lot if the perturbation is small. Running title: Entropy for multidimensional perturbations 2000 Mathematics Subject Classification. Primary 37B40, Secondary 37E05, 37L99.

The fast dynamo growth rate for a C k+1 map or ow is bounded above by topo-logical entropy plus a 1=k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following anti-dynamo theorem: in C 1 systems fast dynamo action is not possible without the presence of chaos. In addition topologica...

Let (X, d) be a compact metric space, f : X 7→ X be a continuous map satisfying a property we call almost specification (which is slightly weaker than the g-almost product property of Pfister and Sullivan), and φ : X 7→ R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topol...

For a countable amenable group Γ and an element f in the integral group ring ZΓ being invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γ on the Pontryagin dual of the quotient of ZΓ by its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f . For the proof, we establish an `-version of Rufus Bowen’s definition of to...

We answer the following question: given any n ∈ N, which is the minimum number of endpoints en of a tree admitting a zero-entropy map f with a periodic orbit of period n? We prove that en = s1s2 ···sk − ∑k i=2 sisi+1 ···sk, where n = s1s2 ···sk is the decomposition of n into a product of primes such that si ≤ si+1 for 1 ≤ i < k. As a corollary, we get a criterion to decide whether a map f defin...

Let (X, T ) be a topological dynamical system (TDS), and h(T, K) the topological entropy of a subset K of X . (X, T ) is lowerable if for each 0 ≤ h ≤ h(T, X) there is a non-empty compact subset with entropy h; is hereditarily lowerable if each non-empty compact subset is lowerable; is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0 ≤ h ≤ h(T, K) there is a no...

We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy o...