Abstract:
For a system of globally coupled chaotic maps with bistable behaviour we relate the rate function for large deviations in the system size at finite time to dynamical properties of the self consistent Perron-Frobenius operator (SCPFO) that describes the system in the infinite size limit.

Abstract:
We discuss how an eigenvalue perturbation formula for transfer operators of dynamical systems is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems. We also list a number of piecewise expanding systems to which this general theory applies and discuss the prospects to apply this theory to some classes of piecewise hyperbolic systems.

Abstract:
We study skew product systems driven by a hyperbolic base map S (e.g. a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on an interval [0,a] like h(x)=g(\theta) tanh(x) where g(\theta) is a factor driven by the base map. The fibre-wise attractor is the graph of an upper semicontinuous function \phi(\theta). For many choices of the function g, \phi has a residual set of zeros but \phi>0 almost everywhere w.r.t. the Sinai-Ruelle-Bowen measure of S^(-1). In such situations we evaluate the stability index of the global attractor of the system, which is the subgraph of \phi, at all regular points (\theta,0) in terms of the local exponents \Gamma(\theta):=\lim_{n\to\infty} 1/n log g_n(\theta) and \Lambda(\theta):=\lim_{n\to\infty} 1/n\log|D_u S^{-n}(\theta)| and of the positive zero s_* of a certain thermodynamic pressure function associated with S^(-1) and g. (In queuing theory, an analogon of s_* is known as Loyne's exponent.) The stability index was introduced by Podvigina and Ashwin in 2011 to quantify the local scaling of basins of attraction.

Abstract:
Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

Abstract:
Let $W_{\lambda,b}(x)=\sum_{n=0}^\infty\lambda^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Bar\'ansky, B\'ar\'any and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda}{\log b}$ for all $\lambda\in(\lambda_b,1)$ with a suitable $\lambda_b<1$. This reproduces results by Bar\'ansky, B\'ar\'any and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.

Abstract:
We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the so called bounded variation approach combined with the ergodic theorem of Ionescu-Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of ``traps'' under the action of random perturbations.

Abstract:
For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue'' means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the SBR invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations. Our main tool in the 1D case is a special technique for ``interchanging'' the map and the perturbation, developed in our previous paper, combined with a compactness argument.

Abstract:
We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{-t}. For a range of parameters two invariant graphs (a trivial and a non-trivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor A_t: Hausdorff and packing dimension have a common value dim(A_t), and there is a critical parameter t_c determined by the SRB measure of T^{-1} such that dim(A_t)=3 for t < t_c and t --> dim(A_t) is strictly decreasing for t_c < t < t_{max}.

Abstract:
We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given.

Abstract:
We introduce a new coupled map lattice model in which the weak interaction takes place via rare "collisions". By "collision" we mean a strong (possibly discontinuous) change in the system. For such models we prove uniqueness of the SRB measure and exponential space-time decay of correlations.