Is there any law that regulates authomatic increases in rent in Germany?
For example, on a rent contract where a certain monthly ammount is established, are there any authomatic yearly increases by law?
No, there are no increases by law. If it increases, it increases because your contract said it increases, or because your contract changed.
There are local laws about trying to reduce the increases of rent, often called "Mietpreisbremse" or "Mietendeckel", but they are local laws and they regulate by how much the rent may be increased. No law exists that defines a default increase or says it must increase.
Increases may happen when the apartment gets modernized or when the area you rent increases (sounds weird at first but for example if your apartment gets an additional balcony, the area you rent has just increased, although your walls haven't moved at all).
What might increase independent of your rent and might actually change by small amounts every year is non-fixed positions, like water, garbage disposal, gardening if you have a common yard, electricity for common rooms (staircase and elevator in an apartment block for example). Those will fluctuate from year to year independent of what is in your contract as rent for your apartment.
Yes, there is but only after a period of 15 months of unchanged rent - whereby raises for renovations (§ 559) and maintainence costs (§ 560) don't count.
§ 558 (1) BGB (Civil Code), generally leads to an automatic increase every 2 years. This increase may be higher than the inflation rate.
These increases lead to average rent of the area being raised, which in itsself leads to further increases.
For pensioners, which recieve about 60% of their previous earnings, this poses problems since these increases alone eats up the increased pension of the last 2 years.
For social support, where the amount of rent is part of the calculation, there is a fix sum of rent that will be taken into consideration. This fix sum is not raised in the same automatic mannor as the rent itself.
Bezahlbarer Wohnraum (Affordable living space) is considered a major social problem where the average rent is nearing the 50% of available income (instead of around 30%), for which § 558 (1) BGB is partially a cause of since it similar in nature to a never ending spiral of increases.