What cognitive properties change as the income vary? Yesterday I made a guess: That perhaps there is a lock in some income classes - that there may be a trend that could get them stuck in there. But it's only a guess, I have no proof for such afirmation.

So are there cognitive differences for people on different income classes (such as rich and poor) that somehow get them stuck in there?

I'm not sure if this question is really on-topic here, but I guess this site could be a first trial. There's also something that I'd like to point out: My native languange is not english, and I've experienced sensitivity when dealing with such matters with inapropriate words: If my words were inapropriate, I didn't mean to be mean, I just don't have a good vocabulary. I also really don't know how to tag this question - If you know better tags, please edit.
Sociology is the study of how we organize ourselves into groups. Common areas of specification in sociology are social stratification, social class, social mobility, religion, secularization, law, and deviance.

Your question specifically is about social mobility which is the movement of peoples between lower, middle and upper classes. It is commonly thought that education allows from vertical social mobility though an achieved status but that isn't accurate. A more advanced educational system than what is currently employed by the university system is necessary to advance socially. 

Now about the specific cognitive difference:


  
   In school, learning tends to be separate from relationships. In the situated learning of poverty, 
  learning occurs in the context of relationships.  
  In school, learning is abstract (represented on paper or computer), verbal (words are relied 
  upon almost exclusively), and proactive (students must plan). In the situated learning 
  environment of poverty, learning is based on sensory data, it relies on nonverbal data as much 
  as verbal, and it is reactive. The two worlds are diametrically opposed.
  In poverty, survival is a crucial skill. In formalized schooling, achievement is the crucial 
  attribute. Survival means the ability to live in the “tyranny of the moment.” It also means one 
  doesn’t develop a future story.  
  School success is highly dependent on a student having an external support system. Often we 
  find that students from poverty are the support system for the family and have very little 
  support for themselves. 
  The intergenerational transfer of knowledge is a huge factor in school success. In a study in 
  Australia that followed more than 8,500 children for 14 years, researchers found they could 
  predict with reasonable accuracy the verbal reasoning scores of 14‐year‐olds based on the 
  maternal grandfather’s occupation (Najman et al., 2004). In other words, the greater the 
  language with young children, the greater their potential (see also No. 7 below). 
  The development of the pre‐frontal cortex, which is the executive function of the brain 
  (impulse control, working memory, planning), generally is not developed by the environment 
  of generational poverty. In a study released in 2008 using EEG (electroencephalogram) scans 
  with poor and middle‐class children, researchers found that the prefrontal cortex of the brain in 
  poor children was undeveloped and resembled the brains of adults who have had strokes 
  (Kishiyama, Boyce, Jimenez, Perry, &amp; Knight, in press). The researchers went on to say that the 
  prefrontal cortex can be developed through intervention. 
  Poverty tends to be heavily dependent upon casual register, whereas school is heavily 
  dependent upon formal register. Hart and Risley (1995) found in their research that the average 
  4‐year‐old in a professional household has heard 45 million words, while a 4‐year‐old in a 
  welfare household has heard 13 million words. In fact, they found that a 4‐year‐old in a 
  professional household has more vocabulary than an adult in a welfare household (Hart &amp; 
  Risley). 
  In any situated learning environment, there is a set of “hidden rules” that individuals tend to 
  follow. Those “hidden rules” are usually not articulated, but they are equated with intelligence.Poverty has a differentset of “hidden rules” frommiddle class and a differentsetfromwealth.
  One set is not better than another; the sets of hidden rules are simply different and help you 
  survive in a given environment.  
  


-How the Environment of Poverty (Having Fewer Resources) Impacts Cognition and Learning 

Stress is thought to be the mechanism by which the brain of poor children is degraded. 


  A massive literature documents the inverse association between poverty or low socioeconomic status and health, but little is known about the mechanisms underlying this robust relation. We examined longitudinal relations between duration of poverty exposure since birth, cumulative risk exposure, and physiological stress in two hundred seven 13-year-olds. Chronic stress was assessed by basal blood pressure and overnight cortisol levels; stress regulation was assessed by cardiovascular reactivity to a standard acute stressor and recovery after exposure to this stressor. Cumulative risk exposure was measured by multiple physical (e.g., substandard housing) and social (e.g., family turmoil) risk factors. The greater the number of years spent living in poverty, the more elevated was overnight cortisol and the more dysregulated was the cardiovascular response (i.e., muted reactivity). Cardiovascular recovery was not affected by duration of poverty exposure. Unlike the duration of poverty exposure, concurrent poverty (i.e., during adolescence) did not affect these physiological stress outcomes. The effects of childhood poverty on stress dysregulation are largely explained by cumulative risk exposure accompanying childhood poverty.


-Childhood poverty and health: cumulative risk exposure and stress dysregulation.