[01000] Advances in random dynamical systems and ergodic theory
Session Date & Time :
01000 (1/3) : 2D (Aug.22, 15:30-17:10)
01000 (2/3) : 2E (Aug.22, 17:40-19:20)
01000 (3/3) : 3C (Aug.23, 13:20-15:00)
Type : Proposal of Minisymposium
Abstract : Random or non-autonomous dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise or seasonal forcing. In recent years, there have been significant advances in the ergodic-theoretic investigation of random dynamical systems, allowing for an enhanced understanding of statistical properties, coherent structures, and the complicated interplay between noise and chaotic dynamics. This minisymposium presents the work of experts and emerging mathematicians working in this vibrant and evolving field, featuring both general-audience lectures giving an overview of the field, and expert-level talks on cutting-edge advances.
Organizer(s) : Alex Blumenthal, Cecilia Gonzalez-Tokman
Cecilia Gonzalez Tokman (University of Queensland)
Jason Atnip (University of Queensland)
Dennis Chemnitz (Free University of Berlin)
Maximilian Engel (Free University of Berlin)
Bernat Bassols-Cornudella (Imperial College London)
Matheus Manzatto de Castro (Imperial College London)
Giuseppe Tenaglia (Imperial College London)
Talks in Minisymposium :
[01338] Random dynamical systems and multiplicative ergodic theorems
Author(s) :
Cecilia Gonzalez Tokman (University of Queensland)
Abstract : Random dynamical systems are flexible mathematical models for the study of complicated systems whose evolution is affected by external factors, such as seasonal cycles and random effects. This talk will start with a broad introduction to the area, with an emphasis on multiplicative ergodic theory. Then, we will review recent advances in the field, which provide fundamental information for the study of transport phenomena in such systems.
[01453] On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape
Author(s) :
Matheus Manzatto de Castro (Imperial College London)
Jeroen S. W. Lamb (Imperial College London)
Martin Rasmussen (Imperial College London)
Vincent P. H. Goverse (Imperial College London)
Abstract : In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu$ on $M$ where the transition kernel $\mathcal{P}$ admits an eigenfunction $0 \leq \eta \in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal{P}$ with respect to $\mu$, which ensure that $\eta(x) \mu(\mathrm{d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure μ-almost surely. We apply this result to the random logistic map $X_{n+1}=\omega_n X_n(1−X_n)$ absorbed at $\mathrm{R}\setminus [0,1]$, where $\omega_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b]$, for $1\leq a < 4$ and $b>4.$
[01767] Lyapunov exponents for random perturbations of coupled standard maps
Author(s) :
Alex Blumenthal (Georgia Tech)
Jinxin Xue (Tsinghua University)
Yun Yang (Virginia Tech)
Abstract : In this talk, we show how to give a quantitative estimate for the sum of the first N Lyapunov exponents for random perturbations of a natural class 2N-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but non invariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both `weak' and `strong' coupling regimes. This is a joint work with Alex Blumenthal and Yun Yang.
[01905] Entropy and pressure formulas for conditioned random dynamical systems
Author(s) :
Maximilian Engel (Free University of Berlin)
Tobias Hurth (Free University of Berlin)
Abstract : Building upon recent results on Lyapunov eponents for memoryless random dynamical systems with absorption --- see Castro et al. 2022 ---, we establish the notion of metric entropy for this setting. We further discuss the relation between entropy, positive Lyapunov exponents and escape rates for such conditioned RDS, where the main example concerns the local dynamics of stochastic differential equations on bounded domains with escape through the boundary. This is joint work with Tobias Hurth.
[02045] Horseshoes for a class of non-uniformly expanding random circle maps
Author(s) :
Giuseppe Tenaglia (Imperial College London)
Abstract : We prove the abundance of horseshoe-like behavior in random circle endomorphisms with a positive Lyapunov exponent under large IID noise conditions. By satisfying a transitivity requirement towards an expanding full branch and effectively controlling the tails of hyperbolic times, our results prove that any two disjoint intervals almost surely exhibit horseshoe-like behavior with full probability.
[05055] Noise-induced chaos and conditioned Lyapunov exponents in a random logistic map
Author(s) :
Bernat Bassols Cornudella (Imperial College London)
Jeroen SW Lamb (Imperial College London)
Abstract : We consider a random logistic map with bounded additive noise, in the parameter regime where the deterministic logistic map has a stable period three cycle. We demonstrate how the transition from noise-induced synchronisation (negative Lyapunov exponent) to noise-induced chaos (positive Lyapunov exponent), arising as a result of growing noise amplitude, can be understood through a two-compartment approximation, effectively modelling the competition between contracting and expanding behaviour. Relevant characteristic exit times and conditioned Lyapunov exponents for the predominantly contracting and expanding compartments are obtained through quasi-stationary and quasi-ergodic invariant measures.
