This paper denes and investigates the ergodic proper-ties of the entropy of a countable partition of a fuzzy dynamical sys-tem at different points of the state space. It ultimately introducesthe local fuzzy entropy of a fuzzy dynamical system and proves itto be an isomorphism invariant.

Extending recent work on stress fluctuations in complex fluids and amorphous solids we describe general terms the ensemble average $$v(\Delta t)$$ standard deviation $$\delta v(\Delta of variance $$v[{\mathbf {x}}]$$ time series $${\mathbf {x}}$$ a stochastic process x(t) measured over finite sampling $$\Delta t$$ . Assuming stationary, Gaussian ergodic process, v$$ is given by functional v_{\m...

If m is Haar measure in G (normalized so that m(G) = 1) we consider the set function m'{E) = m{E). (E is the set of all x, # £ £ . ) Since m is a measure on G possessing all defining properties of m it follows from the uniqueness of Haar measure that ?n'(E) =m(E) for every measurable set E. In other words a is a measure preserving transformation of G; the purpose of this note is to investigate ...

We study a renormalization transformation arising in an infinite system of interacting diffusions. The components of the system are labeled by the N -dimensional hierarchical lattice (N ≥ 2) and take values in a compact convex set D ⊂ Rd (d ≥ 1). Each component starts at some θ ∈ D and is subject to two motions: (1) an isotropic diffusion according to a local diffusion rate g : D → [0,∞) chosen...

We study Abelian sandpiles on graphs of the form G×I, where G is an arbitrary finite connected graph, and I ⊂ Z is a finite interval. We show that for any fixed G with at least two vertices, the stationary measures μI = μG×I have two extremal weak limit points as I ↑ Z. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show ...

Suppose L is a semisimple Levi subgroup of a connected Lie group G, X is a Borel G-space with finite invariant measure, and α : X × G → GLn(R) is a Borel cocycle. Assume L has finite center, and that the real rank of every simple factor of L is at least two. We show that if L is ergodic on X, and the restriction of α to X × L is cohomologous to a homomorphism (modulo a compact group), then, aft...

Let G be a countable discrete amenable group which acts continuously on a compact metric space X and let μ be an ergodic G−invariant Borel probability measure on X. For a fixed tempered Følner sequence {Fn} inG with lim n→+∞ |Fn| logn =∞, we prove the following variational principle: h(Gμ, {Fn}) = hμ(X,G), where Gμ is the set of generic points for μ with respect to {Fn} and h(Gμ, {Fn}) is the B...

We define what it means to “speed up” a Zd−measure-preserving dynamical system, and prove that given any ergodic extension Tσ of a Zd− measure-preserving action by a locally compact, second countable group G, and given any second G−extension Sσ of an aperiodic Zd− measure-preserving action, there is a relative speedup of Tσ which is relatively isomorphic to Sσ . Furthermore, we show that given ...

In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the erg...

We show that a stably ergodic diffeomorphism can be C approximated by a diffeomorphism having stably non-zero Lyapunov exponents. Two central notions in Dynamical Systems are ergodicity and hyperbolicity. In many works showing that certain systems are ergodic, some kind of hyperbolicity (e.g. uniform, non-uniform or partial) is a main ingredient in the proof. In this note the converse direction...