Probabilistic  Methods  in  Chaotic  Dynamics

The  emergence  of  chaotic  systems  and  the  erratic  behavior  that  they  enclose  triggered  a  new  approach  in  their  analysis,  more concerned  with  their  statistical  properties.  In  order  to  learn  about  the  long-­term  behavior  of  these  systems  through  a  probabilistic perspective,  one  can  just  consider  dynamically  defined  stochastic  processes  arising  from  these  systems  by  simply  evaluating  an observable  function  along  the  time  evolution  of  the  orbits  of  the  system.  These  processes  will  be  our  starting  point.  

The  next  step  is  to  find  invariant  ergodic  probability  measures  with  physical  relevance.  The  time  invariance  of  these  measures implies  that  the  dynamically  defined  stochastic  processes  become  stationary  with  respect  to  these  measures.  The  physical  relevance derives  from  the  fact  that  we  require  that  the  law  of  large  numbers  holds  for  a  set  of  initial  conditions  with  positive  volume.  These measures  are  usually  called  Sinal-­Ruelle-­Bowen  (SRB)  measures,  authors  who  in  their  remarkable  works  established  their  existence for  uniformly  hyperbolic  dynamical  systems.  Moreover,  besides  providing  a  description  of  time  averages,  SRB  measures  can  be identified  as  equilibrium  states  for  certain  chosen  potentials.  One  of  our  main  problems  consists  in  studying  the  existence  of  SRB measures  for  dynamical  systems  exhibiting  weak  forms  of  hyperbolicity.

While  SRB  measures  immediately  supply  laws  of  large  numbers  for  dynamically  defined  stochastic  processes,  for  other  finer statistical  properties,  ergodicity  is  not  enough  and  some  stronger  feature  like  mixing  is  needed.  Rates  of  mixing,  which  establish how  fast  the  system  looses  memory,  are  usually  written  (proved)  in  terms  of  decay  of  correlations.  Sufficiently  fast  decay  of correlations  can  be  used  to  obtain  central  limit  theorems  or  rates  for  large  deviations.  In  the  last  two  decades  there  were  many significant  advances  in  the  non-­uniformly  hyperbolic  context,  but  some  interesting  questions  in  this  field  still  remain  open.  

Most  of  the  statistical  properties  considered  so  far  pertain  to  the  mean  or  average  behavior  of  the  orbits.  However,  in  certain circumstances  like  in  risk  assessment,  one  is  more  interested  in  extreme  observations  (exceeding  high  thresholds)  that  correspond to  the  occurrence  of  rare  events.  Extreme  value  laws  are  related  to  the  hitting  times  statistics  or  return  times statistics.  We  aim  at  giving  contributions  on  this  topic  for  slowly  mixing  systems  and  systems  with  sigma-­finite  measures.

Another  important  issue  concerns  the  stability  of  systems.  Despite  of  remarkable  successes  in  uniformly  hyperbolic  systems, structural  stability  proved  to  be  too  strong  a  requirement  for  many  applications,  in  particular,  among  some  classes  of  chaotic dynamical  systems.  More  recently,  increasing  emphasis  has  been  put  on  expressing  stability  in  terms  of  persistence  of  statistical properties  of  the  system.  A  natural  formulation,  the  one  that  concerns  us  most  in  this  work,  corresponds  to  continuous  variation  of physical  measures  as  a  function  of  the  dynamical  system,  where  in  the  space  of  probability  measures  we  consider  the  weak* topology  or,  when  it  makes  sense,  the  L^1  norm  in  the  space  of  densities  with  respect  to  Lebesgue  measure.  

A  fruitful  way  of  facing  chaotic  dynamical  systems  is  through  random  perturbations  and  stationary  measures.  We  say  that  the original  map  is  stochastically  stable  if  the  stationary  measures  converge  (in  the  weak*  topology  or  L^1  norm,  if  possible)  to  the  SRB measure  of  the  original  map  when  the  support  of  the  probabilistic  law  giving  the  possible  choices  of  iterations  shrinks  to  the  original map.  Stochastic  stability  tries  to  reflect  that  the  introduction  of  small  random  noise  affects  just  slightly  the  statistical  description  of the  dynamical  system  and  it  has  been  proved  in  great  generality for  non-­uniformly  expanding  maps.  One  of  our  goals  is to  obtain  stochastic  stability  for  other  classes  of  maps  and  deduce  some  properties  of  the  stationary  measures  as  well.

Finally,  another  of  our  goals  is  to  understand  mechanisms  leading  to  the  creation  of  chaotic  dynamical  systems.  Changes  in  the dynamics  that  lead  to  the  creation  of  strange  phenomena  is  always  a  key  step  to  obtain  new  general  results.  In  our  specific  case, we  will  be  particularly  interested  in  bifurcations  via  heterodimensional  cycles  and  the  dynamics  arising  from  heterodimensional cycles  for  diffeomorphisms,  which  on  their  own  motivate  the  study  of  iterated  function  systems.

Another  way  to  obtain  chaotic  behavior  may  be  by  introducing  an  impulse  in  any  region  in  the  phase  space.  The  theory  of  impulsive dynamical  systems  comes  from  the  impulsive  differential  equations  and,  so  far,  the  most  significant  part  of  its  development  has been  focused  on  geometric/topological  problems  related  to  the  existence  and  uniqueness  of  solutions  and  characterization  of  the limit  sets.  A  first  result  on  the  statistical  properties  of  impulsive  dynamical  systems established sufficient  conditions  for  the  existence  of  invariant  probability  measures.  In  the  wake  of  this  result,  many  interesting  questions  arise towards  an  ergodic  theory  of  impulsive  dynamical  systems,  such  as  the  validity  of  a  Variational  Principle,  the  existence  of  maximum entropy  measures,  the  existence  of  physical  measures,  etc.



Start date: 

Sunday, 1 May, 2016

Area / Group: 

Dynamical Systems


Ana Cristina Moreira Freitas
Fernando Jorge Soares Moreira
Jorge Milhazes Freitas
Jorge Rocha
José Ferreira Alves
Maria Carvalho

Other members: 

David Mesquita
Helder Vilarinho
Jaqueline Rocha
Lorenzo J. Díaz
Marcelo Viana
Mike Todd
Muhammad Ali Khan
Sandro Vaienti
Stefano Luzzatto
Vilton Pinheiro

Financial support: 

151 855EUR